Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis

Abstract

Recently, Denniston, Melton, and Rodabaugh presented a new categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille; their outlook was based on the notion of lattice-valued interchange system and a category of Galois connections. This paper extends the approach of Denniston et al. clarifying the relationships between Chu spaces of Pratt, many-valued formal contexts of FCA, lattice-valued interchange systems, and Galois connections.

1. Introduction

This paper considers a particular application of the theory of variety-based topological systems, introduced in [1] as a generalization of topological systems of Vickers [2], which in turn provide a common framework for both topological spaces and their underlying algebraic structures frames (also called locales), thereby allowing researchers to switch freely between the spatial and localic viewpoints. Vickers’ concept, which was motivated by the theory of crisp topology, has recently drawn the attention of fuzzy topologists, who have incorporated the notion in their topic of study. The first attempt in this respect was done by Denniston and Rodabaugh [3], who considered various functorial relationships between topological systems of Vickers and lattice-valued topological spaces of Rodabaugh [4]. Soon afterwards, the study of Guido [5, 6] followed; he used topological systems in order to construct a functor from the category of 𝐿-topological spaces to the category of crisp topological spaces, thereby providing a common framework for various approaches to the hypergraph functor of the fuzzy community (see, e.g., [7]).

It soon appeared though that the translation of the most important tools of the theory of topological systems (e.g., system spatialization and localification procedures) into the language of lattice-valued topology required a suitable fuzzification of the former concept, which was accomplished successfully by Denniston et al. in [8]. The authors brought their theory into maturity in [9], where they considered its applications to both lattice-valued variable-basis topology of Rodabaugh [4] and (𝐿,𝑀)-fuzzy topology of Kubiak and Šostak [10]. Later on, they introduced a particular instance of their concept called interchange system [11], which was motivated by certain aspects of program semantics (the so-called predicate transformers) initiated by Dijkstra [12]. It should be noted, however, that an interchange system is a particular instance of the well-known concept of Chu space in the sense of Pratt [13] (the original notion goes back to Barr [14]). Moreover, the interchange system morphisms of [11] are precisely the Chu space morphisms (called Chu transforms) of [13]. (In [15], Denniston et al. use the terms transformer system and transformer morphism.) The underlying motivations of interchange systems and Chu spaces are quite different, which is reflected well enough in both their names and their respective theories.

Parallel to the previously mentioned developments, we introduced the concept of variety-based topological system [1], which not only extended the setting of Denniston et al., but also included the case of state property systems of Aerts [16–18] (introduced as the basic mathematical structure in the Geneva-Brussels approach to foundations of physics), considered in [19] in full detail, and we brought the functor of Guido to a new level [20] (e.g., made it variable-basis in the sense of Rodabaugh [4], as well as extended the machinery of Höhle [7], to construct its right adjoint functor). Moreover, in [21], we provided a thorough categorical extension of the system spatialization procedure to the variety-based setting, the respective localification procedure being elaborated in [22]. Additionally, certain aspects of the new theory bearing links to bitopological spaces of Kelly [23] as well as to noncommutative topology of Mulvey and Pelletier [24, 25] were treated extensively in [26, 27], respectively.

Seeing the fruitfulness of their newly introduced notions, both Denniston et al. and the author of this paper independently turned their attention to possible applications of the arising system framework to the areas beyond topology. In particular, in [28], we started the theory of lattice-valued soft universal algebra, whereas Denniston et al. presented in [29, 30] a challenging categorical outlook on a certain lattice-valued extension of Formal Concept Analysis (FCA) of Ganter and Wille [31]. Similar to the case of lattice-valued topological systems, their research motivated us to take a deeper look into the proposed topic of lattice-valued FCA, thereby streamlining the motivating approach of Denniston et al. and also extending some of their results.

Modifying slightly the definition of the category of Galois connections of McDill et al. [32], Denniston et al. [29] consider its relationships to the already mentioned category of interchange systems, the objects of both categories being essentially formal contexts of FCA. The main result is a categorical embedding of each category into its counterpart, the ultimate conclusion though amounting to the categories in question being rather different; that is, the two viewpoints on FCA are not categorically isomorphic. The rest of the paper provides a lattice-valued, fixed-basis extension of the crisp achievements, based on the lattice-valued FCA of Bêlohlávek [33]. Moreover, towards the end of the paper, with the help of the constructed embeddings, the authors compare the categories of (lattice-valued) formal contexts and (lattice-valued) topological systems inside the category of (lattice-valued) interchange systems, making a significant (and rather striking) metamathematical conclusion on their being disjoint.

The just mentioned approach of Denniston et al. can be extended, taking into consideration the following facts. Firstly, it never gives much attention to the links between interchange systems and Chu spaces, thereby making no use of the well-developed machinery of the latter. Secondly, while pursuing their approach to many-valued FCA, the authors never mention the already existing many-valued formal contexts of Ganter and Wille [31], which (being similar to Chu spaces) essentially extend their own. Thirdly, being categorically oriented, the paper introduces two types of formal context morphisms, in terms of Chu transforms and Galois connections, respectively, but misses the well-established context morphisms of FCA, which are different from those of the paper. Lastly (and most important), turning to fuzzification of their notions, the authors concentrate on the fixed-basis lattice-valued approach, notwithstanding that the more promising variable-basis setting (initiated by one of them) has already gained its popularity in the fuzzy community, currently being a kind of de facto standard for fuzzy structures.

The main purposes of this paper are to extend the results of Denniston et al. and to clarify the relationships between Chu spaces, many-valued formal contexts of FCA, lattice-valued interchange systems, and Galois connections. In particular, we introduce three categories (each of them having a twofold modification, arising from its respective variable-basis), whose objects are lattice-valued versions of formal contexts of FCA, but whose morphisms are quite different, each instance reflecting the approach of Pratt, Ganter and Wille, and Denniston et al., respectively. Following the results of [29], we embed (nonfully) each of the categories into its two counterparts, showing, however, that the constructed embeddings fail to provide a categorical isomorphism (even an adjoint situation) between any two of the presented three approaches. As a consequence, we obtain three different categorical outlooks on FCA, which are done in the variable-basis lattice-valued framework of [4]. Moreover, the respective underlying lattices of the fuzzified formal contexts are quantales [34, 35] instead of complete residuated lattices of Denniston et al.; that is, the heavy assumption on commutativity of the quantale multiplication is dropped. In particular, we show lattice-valued generalizations of formal concept, preconcept, and protoconcept of [36], which are based on an arbitrary (namely, non-commutative) quantale (actually, an algebra, having a quantale as its reduct).

Similar to Denniston et al., we develop the required machinery of quantale-valued Galois connections, which extends that of Erné et al. [37] and which is motivated by the commutative approach of fuzzy Galois connections of Bêlohlávek [38]. For example, we provide yet another generalization to the quantale setting of the well-known fact that every order-reversing Galois connection between powersets arises as a pair of Birkhoff operators with respect to a binary relation [39]. The generalization though appears to be somewhat truncated, in the sense that while lattice-valued binary relations still give rise to order-reversing Galois connections, the converse way is not always available ([38, 40] do restore the full result even in the lattice-valued case, but rely on the more demanding notion of fuzzy Galois connection). Moreover, we define a category of order-preserving Galois connections as a certain subcategory of the category of Chu spaces, in which (unlike McDill et al. and Denniston et al.) Galois connections play the role of morphisms rather than objects, thereby providing particular instances of Chu transforms. We fail to provide a similar machinery for the order-reversing case.

At the end of the paper, we will notice that the approach to FCA through order-reversing Galois connections taken up by Denniston et al. does not suit exactly the underlying ideas of FCA of Ganter and Wille, which are formulated in terms of the incidence relations of formal contexts. The main sticking point here is the previously mentioned fact that the important crisp case one-to-one correspondence between order-reversing Galois connections on powersets and pairs of Birkhoff operators breaks down in the lattice-valued case (quantales). As a consequence, the incidence relation viewpoint provides one with a Galois connection, whereas the latter alone lacks the strength of bringing a respective incidence relation in play (cf. Problems 1 and 2).

The paper is based on both category theory and universal algebra, relying more on the former. The necessary categorical background can be found in [41–43]. For algebraic notions, the reader is referred to [34, 35, 44, 45]. Although we tried to make the paper as much self-contained as possible, it is expected from the reader to be acquainted with basic concepts of category theory, that is, those of category and functor.

2. Algebraic Preliminaries

For convenience of the reader, we begin with those algebraic preliminaries, which are crucial for the understanding of the results of the paper. Experienced researchers may skip this section, consulting it for the notations of the author only.

2.1. Varieties of Algebras

An important foundational aspect of the paper is the abstract algebraic structure (called algebra, for short) which is a set, equipped with a family of operations, satisfying certain identities. The theory of universal algebra calls a class of finitary algebras (induced by a set of finitary operations), which is closed under the formation of homomorphic images, subalgebras, and direct products, a variety. In this paper, we extend the notion of varieties to include infinitary algebraic theories. Our motivation includes ideas of [45–47].

Definition 1. Let 𝛀=(𝑛_𝜆)_𝜆∈Λ be a (possibly, proper or empty) class of cardinal numbers. An 𝛀-algebra is a pair (𝐴,(𝜔^𝐴_𝜆)_𝜆∈Λ), which comprises a set 𝐴 and a family of maps 𝐴−−→^{𝑛𝜆𝜔𝐴𝜆}𝐴 (𝑛_𝜆-ary primitive operations on 𝐴). An 𝛀-homomorphism (𝐴₁,(𝜔^𝐴1_𝜆)_𝜆∈Λ)→−^𝜑(𝐴₂,(𝜔^𝐴2_𝜆)_𝜆∈Λ) is a map 𝐴→−₁^𝜑𝐴₂ such that the diagram (1) commutes for every 𝜆∈Λ. 𝐀𝐥𝐠(𝛀) is the construct of 𝛀-algebras and 𝛀-homomorphisms.

Every concrete category of this paper has an underlying functor |−| to the respective ground category (e.g., the category Set of sets and maps), the latter mentioned explicitly in each case.

Definition 2. Let ℳ (resp., ℰ) be the class of 𝛀-homomorphisms with injective (resp., surjective) underlying maps. A variety of 𝛀-algebras are a full subcategory of 𝐀𝐥𝐠(𝛀), closed under the formation of products, ℳ-subobjects (subalgebras), and ℰ-quotients (homomorphic images). The objects (resp., morphisms) of a variety are called algebras (resp., homomorphisms).

Definition 3. Given a variety 𝐀, a reduct of 𝐀 is a pair (‖−‖,𝐁), where 𝐁 is a variety such that 𝛀_𝐁⊆𝛀_𝐀, and 𝐀−−−→^‖−‖𝐁 is a concrete functor. The pair (𝐀,‖−‖) is called then an extension of 𝐁.

The following definitions recall several well-known varieties [34, 35, 48, 49], thereby providing the background for the subsequent developments of this paper.

Definition 4. 𝐂𝐒𝐋𝐚𝐭(Ξ) is the variety of Ξ-semilattices, that is, partially ordered sets (posets), which have arbitrary Ξ∈{⋀,⋁} (meets and joins, resp.).

Given a Ξ-semilattice 𝐴, we have both ⋀ and ⋁ (however, only one of them is respected by Ξ-semilattice homomorphisms) and denote its largest (resp., smallest) element ⊤_𝐴 (resp., ⟂_𝐴). Given a ⋁-semilattice 𝐴, we denote ≀𝐴≀ its respective ⋀-semilattice and vice versa. The standard result of [50, Sections 0--3] says that every ⋁-semilattice homomorphism 𝐴→−₁^𝜑𝐴₂ has the upper adjoint map |𝐴₂|−−→^𝜑⊢|𝐴₁|, which is characterized uniquely by the condition 𝜑(𝑎₁)⩽𝑎₂ iff 𝑎₁⩽𝜑^⊢(𝑎₂) for every 𝑎₁∈𝐴₁ and every 𝑎₂∈𝐴₂, which, for order-preserving maps only, is equivalent to the properties 𝜑∘𝜑^⊢⩽1_𝐴2 and 1_𝐴1⩽𝜑^⊢∘𝜑 (given an algebra 𝐴, 1_𝐴 denotes the identity map on 𝐴). The explicit formula for the map is given by 𝜑^⊢(𝑎₂)=⋁{𝑎₁∈𝐴₁|𝜑(𝑎₁)⩽𝑎₂}, whereas the map itself is easily shown to be ⋀-preserving. The dual description of the lower adjoint map (based on a ⋀-semilattice homomorphism 𝐴→−₁^𝜑𝐴₂ and denoted by |𝐴₂|−−→^𝜑⊣|𝐴₁|) is left to the reader (see also Section 2.3 on Galois connections).

Definition 5. 𝐐𝐮𝐚𝐧𝐭 is the variety of quantales, that is, triples (𝑄,⋁,⊗) where(1)(𝑄,⋁) is a ⋁-semilattice; (2)(𝑄,⊗) is a semigroup; that is, 𝑞₁⊗(𝑞₂⊗𝑞₃)=(𝑞₁⊗𝑞₂)⊗𝑞₃ for every 𝑞₁,𝑞₂,𝑞₃∈𝑄;(3)⊗ distributes across ⋁ from both sides; that is, 𝑞⊗(⋁𝑆)=⋁_𝑠∈𝑆(𝑞⊗𝑠) and (⋁𝑆)⊗𝑞=⋁_𝑠∈𝑆(𝑠⊗𝑞) for every 𝑞∈𝑄 and every 𝑆⊆𝑄.

Every quantale 𝑄 has two residuations, which are induced by its binary operation ⊗ (called multiplication) and which are defined by 𝑞₁→_𝑙𝑞₂=⋁{𝑞∈𝑄|𝑞⊗𝑞₁⩽𝑞₂} and 𝑞₁→_𝑟𝑞₂=⋁{𝑞∈𝑄|𝑞₁⊗𝑞⩽𝑞₂} (notice that “𝑙" (resp., “𝑟") stands for “multiply on the left" (resp., “right”)), respectively, providing a single residuation ⋅→⋅ in case of a commutative multiplication (resulting in complete residuated lattices of Denniston et al. [29]). These operations have the standard properties of poset adjunctions [50, Sections 0--3] or (order-preserving) Galois connections [37] (see Section 2.3 of this paper), for example, 𝑞₁⩽𝑞₂→_𝑙𝑞₃ iff 𝑞₁⊗𝑞₂⩽𝑞₃ iff 𝑞₂⩽𝑞₁→_𝑟𝑞₃. For convenience of the reader, the following theorem recalls some of their other features, which will be heavily used throughout the paper.

Theorem 6. The residuations ⋅→_𝑙⋅ and ⋅→_𝑟⋅ of a quantale 𝑄 have the following properties.(1)𝑞→_𝑙(⋀𝑆)=⋀_𝑠∈𝑆(𝑞→_𝑙𝑠) and 𝑞→_𝑟(⋀𝑆)=⋀_𝑠∈𝑆(𝑞→_𝑟𝑠) for every 𝑞∈𝑄 and every 𝑆⊆𝑄. In particular, 𝑞→_𝑙⊤_𝑄=⊤_𝑄 and 𝑞→_𝑟⊤_𝑄=⊤_𝑄 for every 𝑞∈𝑄. (2)(⋁𝑆)→_𝑙𝑞=⋀_𝑠∈𝑆(𝑠→_𝑙𝑞) and (⋁𝑆)→_𝑟𝑞=⋀_𝑠∈𝑆(𝑠→_𝑟𝑞) for every 𝑞∈𝑄 and every 𝑆⊆𝑄. In particular, ⟂_𝑄→_𝑙𝑞=⊤_𝑄 and ⟂_𝑄→_𝑟𝑞=⊤_𝑄 for every 𝑞∈𝑄. (3)𝑞₁⩽(𝑞₁→_𝑙𝑞₂)→_𝑟𝑞₂ and 𝑞₁⩽(𝑞₁→_𝑟𝑞₂)→_l𝑞₂ for every 𝑞₁,𝑞₂∈𝑄. (4)(𝑞₁⊗𝑞₂)→_𝑙𝑞₃=𝑞₁→_𝑙(𝑞₂→_l𝑞₃) and (𝑞₁⊗𝑞₂)→_𝑟𝑞₃=𝑞₂→_𝑟(𝑞₁→_𝑟𝑞₃) for every 𝑞₁,𝑞₂,𝑞₃∈𝑄.

Proof. To give the flavor of the employed machinery, we show the proofs of some of the claims of the items. Ad (3). 𝑞₁→_𝑙𝑞₂⩽𝑞₁→_𝑙𝑞₂ implies (𝑞₁→_𝑙𝑞₂)⊗𝑞₁⩽𝑞₂ implies 𝑞₁⩽(𝑞₁→_𝑙𝑞₂)→_𝑟𝑞₂. Ad (4). Given 𝑞∈𝑄, 𝑞⩽(𝑞₁⊗𝑞₂)→_𝑙𝑞₃ iff 𝑞⊗(𝑞₁⊗𝑞₂)⩽𝑞₃ iff 𝑞⊗𝑞₁⩽𝑞₂→_𝑙𝑞₃ iff 𝑞⩽𝑞₁→_𝑙(𝑞₂→_𝑙𝑞₃).
Moreover, 𝑞⩽(𝑞₁⊗𝑞₂)→_𝑟𝑞₃ iff (𝑞₁⊗𝑞₂)⊗𝑞⩽𝑞₃ iff 𝑞₂⊗𝑞⩽𝑞₁→_𝑟𝑞₃ iff 𝑞⩽𝑞₂→_𝑟(𝑞₁→_𝑟𝑞₃).

In the subsequent developments, we will consider some special types of quantales mentioned in the following.

Definition 7. CQuant is the variety of commutative quantales, which are quantales 𝑄 with commutative multiplication; that is, 𝑞₁⊗𝑞₂=𝑞₂⊗𝑞₁ for every 𝑞₁, 𝑞₂∈𝑄.

Definition 8. UQuant is the variety of unital quantales, which are quantales 𝑄 having an element 1_𝑄 such that the triple (𝑄,⊗,1_𝑄) is a monoid; that is, 𝑞⊗1_𝑄=𝑞=1_𝑄⊗𝑞 for every 𝑞∈𝑄.

Definition 9. SRSQuant (resp., SLSQuant) is the variety of strictly right-sided (resp., left-sided) quantales, which are quantales 𝑄 such that 𝑞⊗⊤_𝑄=𝑞 (resp., ⊤_𝑄⊗𝑞=𝑞) for every 𝑞∈𝑄. STSQuant is the variety of strictly two-sided quantales, that is, unital quantales whose unit is the top element.

Definition 10. Frm is the variety of frames, that is, strictly two-sided quantales, in which multiplication is the meet operation.

Definition 11. CBAlg is the variety of complete Boolean algebras, that is, frames 𝐴, in which ⋀ are considered as primitive operations (cf. Definition 1) and which additionally are equipped with the complement operation (−)^′, assigning to every 𝑎∈𝐴 an element 𝑎^′∈𝐴 such that 𝑎∧𝑎^′=⟂_𝐴 and 𝑎∨𝑎^′=⊤_𝐴.

The varieties CBAlg, Frm, STSQuant, UQuant, and 𝐂𝐒𝐋𝐚𝐭(⋁) provide a sequence of variety reducts in the sense of Definition 3. Additionally, Frm is an extension of CQuant.

The notations related to varieties in this paper follow the category-theoretic pattern (which is different from the respective one of the fuzzy community). From now on, varieties are denoted by A or B, whereas those which extend the variety 𝐂𝐒𝐋𝐚𝐭(Ξ) of Ξ-semilattices with Ξ∈{⋁,⋀} will be distinguished with the notation L. S will stand for either subcategories of varieties or the subcategories of their dual categories. The categorical dual of a variety 𝐀 is denoted by 𝐀^𝑜𝑝. However, the dual of Frm uses the already accepted notation Loc [48]. Given a homomorphism 𝜑, the corresponding morphism of the dual category is denoted by 𝜑^𝑜𝑝 and vice versa. Every algebra 𝐴 of a variety 𝐀 gives rise to the subcategory 𝐒_𝐴 of either 𝐀 or 𝐀^𝑜𝑝, whose only morphism is the identity map 𝐴−−→^1𝐴𝐴.

2.2. Powerset Operators

One of the main tools in the subsequent developments of the paper will be generalized versions of the so-called forward and backward powerset operators. More precisely, given a map 𝑋−→₁^𝑓𝑋₂, there exists the forward (resp., backward) powerset operator 𝒫(𝑋₁)−−−→^𝑓→𝒫(𝑋₂) (resp., 𝒫(𝑋₂)−−−→^𝑓←𝒫(𝑋₁)) defined by 𝑓^→(𝑆)={𝑓(𝑠)|𝑠∈𝑆} (resp., 𝑓^←(𝑇)={𝑥∈𝑋₁|𝑓(𝑥)∈𝑇}). The two operators admit functorial extensions to varieties of algebras. (For a thorough treatment of the powerset operator theory, the reader is referred to the articles of Rodabaugh [49, 51–53], dealing with the lattice-valued case, or to the papers of the author himself [21, 54], dealing with a more general variety-based approach.)

To begin with, notice that given an algebra 𝐴 of a variety 𝐀 and a set 𝑋, there exists the 𝐴-powerset of 𝑋, namely, the set 𝐴^𝑋 of all maps 𝑋→−^𝛼𝐴. Equipped with the pointwise algebraic structure involving that of 𝐴, 𝐴^𝑋 provides an algebra of 𝐀. In the paper, we will often use specific elements of this powerset algebra. More precisely, every 𝑎∈𝐴 gives rise to the constant map 𝑋→−^𝑎𝐴 with the value 𝑎. Moreover, if 𝐀 extends the variety 𝐂𝐒𝐋𝐚𝐭(⋁) of ⋁-semilattices, then every 𝑆⊆𝑋 and every 𝑎∈𝐴 give rise to the map 𝑋−−→^𝜒𝑎𝑆𝐴 (𝑎-characteristic map of 𝑆), which is defined by 𝜒^𝑎_𝑆(𝑥)=⟂𝑎,𝑥∈𝑆_𝐴,otherwise. (2) Of particular importance will be the characteristic maps of singletons, that is, 𝜒^⊤𝐴_{𝑥} or 𝜒^1𝐴_{𝑥}, the latter case assuming additionally that 𝐀 extends the variety UQuant of unital quantales.

Coming back to powerset operators, the following extension of the forward one has already appeared in a more general way in [28], being different from the respective lattice-valued approaches of Rodabaugh.

Theorem 12. Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(⋁), every subcategory 𝐒 of 𝐋 provides the functor 𝐒𝐞𝐭×𝐒−−−−→^(−)→𝐂𝐒𝐋𝐚𝐭(⋁), which is defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^→=‖𝐿^𝑋1₁‖−−−−−−→^{(𝑓,𝜑)→}‖𝐿^𝑋2₂‖, where ((𝑓,𝜑)^→(𝛼))(𝑥₂)=𝜑(⋁_{𝑓(𝑥1)=𝑥2}𝛼(𝑥₁)).

Proof. To show that ‖𝐿^𝑋1₁‖−−−−−−→^{(𝑓,𝜑)→}‖𝐿^𝑋2₂‖ is ⋁-preserving, notice that given 𝑆⊆𝐿^𝑋1₁, for every 𝑥₂∈𝑋₂, ((𝑓,𝜑)^→(⋁𝑆))(𝑥₂)=𝜑(⋁_{𝑓(𝑥1)=𝑥2}(⋁𝑆)(𝑥₁))=𝜑(⋁_{𝑓(𝑥1)=𝑥2}⋁_𝑠∈𝑆𝑠(𝑥₁))=𝜑(⋁_𝑠∈𝑆⋁_{𝑓(𝑥1)=𝑥2}𝑠(𝑥₁⋁))=_𝑠∈𝑆𝜑(⋁_{𝑓(𝑥1)=𝑥2}𝑠(𝑥₁))=⋁_𝑠∈𝑆((𝑓,𝜑)^→(𝑠))(𝑥₂)=(⋁_𝑠∈𝑆(𝑓,𝜑)^→(𝑠))(𝑥₂).
To show that the functor preserves composition, notice that given 𝐒𝐞𝐭×𝐒-morphisms (𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂) and (𝑋₂,𝐿₂)−−−−−→^{(𝑓′,𝜑′)}(𝑋₃,𝐿₃), for every 𝛼∈𝐿^𝑋1₁ and every 𝑥₃∈𝑋₃, ((𝑓^′,𝜑^′)^→∘(𝑓,𝜑)^→(𝛼))(𝑥₃)=𝜑^′(⋁_{𝑓′(𝑥2)=𝑥3}((𝑓,𝜑)^→(𝛼))(𝑥₂))=𝜑^′(⋁_{𝑓′(𝑥2)=𝑥3}𝜑(⋁_{𝑓(𝑥1)=𝑥2}𝛼(𝑥₁)))=𝜑^′∘𝜑(⋁_{𝑓′∘𝑓(𝑥1)=𝑥3}𝛼(𝑥₁))=((𝑓^′∘𝑓,𝜑^′∘𝜑)^→(𝛼))(𝑥₃)=(((𝑓^′,𝜑^′)∘(𝑓,𝜑))^→(𝛼))(𝑥₃).

The case 𝐒=𝐒_𝐿 is denoted by (−)^→_𝐿 and is called a (variety-based) fixed-basis approach, whereas all other cases are subsumed under (variety-based) variable-basis approach. Similar notations and naming conventions are applicable to all other variety-based extensions of powerset operators introduced in this subsection and, for the sake of brevity, will not be mentioned explicitly.

There is another extension of the forward powerset operator, which uses the idea of Rodabaugh [52] (the reader is advised to recall the notation for lower and upper adjoint maps from the previous subsection). Before the actual result, however, some words are due to our employed powerset operator notations. In all cases, which are direct analogues of the classical powerset operators of the fuzzy community (e.g., those of Rodabaugh [51] and his followers), we use the solid arrow notation, that is, “(−)^←” (resp., “(−)^→”) for backward (resp., forward) powerset operator. In the cases, which are not well known to the researchers, we use dashed arrow notation, that is, “(−)^⇠” (resp., “(−)^⇢") for backward (resp., forward) powerset operator, possibly, adding some new symbols (e.g., “⊣” or “⊢”) to underline the involvement of upper or lower adjoint maps in the definition of the operator in question (e.g., (−)^⊣⇢ or (−)^⊢⇢) as in the following theorem.

Theorem 13. (1) Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(⋀), every subcategory 𝐒 of 𝐋^𝑜𝑝 gives rise to the functor 𝐒𝐞𝐭×𝐒−−−−−→^(−)⊣⇢𝐂𝐒𝐋𝐚𝐭(⋁) defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^⊣⇢=≀𝐿^𝑋1₁≀−−−−−−→^{(𝑓,𝜑)⊣⇢}≀𝐿^𝑋2₂≀, where ((𝑓,𝜑)^⊣⇢(𝛼))(𝑥₂)=𝜑^𝑜𝑝⊣(⋁_{𝑓(𝑥1)=𝑥2}𝛼(𝑥₁)). (2) Let 𝐋 be a variety, which extends 𝐂𝐒𝐋𝐚𝐭(⋁), and let 𝐒 be a subcategory of 𝐋^𝑜𝑝 such that for every 𝐒-morphism 𝐿→−₁^𝜑𝐿₂, the map |𝐿₁|−−−→^{𝜑𝑜𝑝⊢}|𝐿₂| is ⋁-preserving. Then there exists the functor 𝐒𝐞𝐭×𝐒−−−−−→^(−)⊢⇢𝐂𝐒𝐋𝐚𝐭(⋁) defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^⊢⇢=‖𝐿^𝑋1₁‖−−−−−−→^{(𝑓,𝜑)⊢⇢}‖𝐿^𝑋2₂‖, where ((𝑓,𝜑)^⊢⇢(𝛼))(𝑥₂)=𝜑^𝑜𝑝⊢(⋁_{𝑓(𝑥1)=𝑥2}𝛼(𝑥₁)).

Proof. We rely on the fact that every ⋀- (resp., ⋁-) semilattice is actually a complete lattice and, therefore, is a ⋁- (resp., ⋀-) semilattice. The proof then follows the steps of those of Theorem 12, employing the fact that given 𝐂𝐒𝐋𝐚𝐭(⋀)- (resp., 𝐂𝐒𝐋𝐚𝐭(⋁)-) homomorphisms 𝐿→−₁^𝜑𝐿₂ and 𝐿−−→₂^𝜑′𝐿₃, it follows that (𝜑^′∘𝜑)^⊣=𝜑^⊣∘𝜑^′⊣ (resp., (𝜑^′∘𝜑)^⊢=𝜑^⊢∘𝜑^′⊢).

The respective generalization of the backward powerset operator is even stronger (see, e.g., [21]), in the sense that the employed varieties do not need to be related to any kind of lattice-theoretic structure.

Theorem 14. Given a variety 𝐀, every subcategory 𝐒 of 𝐀^𝑜𝑝 induces the functor 𝐒𝐞𝐭×𝐒−−−−→^(−)←𝐀^𝑜𝑝, which is defined by ((𝑋₁,𝐴₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐴₂))^←=𝐴−−−−−−−−→^𝑋1₁^{((𝑓,𝜑)←)𝑜𝑝}𝐴^𝑋2₂, where (𝑓,𝜑)^←(𝛼)=𝜑^𝑜𝑝∘𝛼∘𝑓.

Proof. To show that 𝐴−−−−−→^𝑋2₂^{(𝑓,𝜑)←}𝐴^𝑋1₁ is an 𝐀-homomorphism, notice that given 𝜆∈Λ_𝐀 and ⟨𝛼_𝑖⟩_𝑛𝜆∈(𝐴^𝑋2₂)^𝑛𝜆, for every 𝑥₁∈𝑋₁, it follows that 𝑓,𝜑^←𝜔^𝐴𝑋22_𝜆𝛼_𝑖𝑛𝜆𝑥₁=𝜑^𝑜𝑝∘𝜔^𝐴𝑋22_𝜆𝛼_𝑖𝑛𝜆∘𝑓𝑥₁=𝜑^𝑜𝑝∘𝜔^𝐴2_𝜆𝛼_𝑖∘𝑓𝑥_1𝑛𝜆=𝜔^𝐴1_𝜆𝜑^𝑜𝑝∘𝛼_𝑖∘𝑓𝑥_1𝑛𝜆=𝜔^𝐴1_𝜆𝑓,𝜑^←𝛼_𝑖𝑥_1𝑛𝜆=𝜔^𝐴𝑋11_𝜆𝑓,𝜑^←𝛼_𝑖𝑛𝜆𝑥₁. (3)
To show that the functor preserves composition, notice that given 𝐒𝐞𝐭×𝐒-morphisms (𝑋₁,𝐴₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐴₂) and (𝑋₂,𝐴₂)−−−−−→^{(𝑓′,𝜑′)}(𝑋₃,𝐴₃), for every 𝛼∈𝐴^𝑋3₃, (𝑓,𝜑)^←∘(𝑓^′,𝜑^′)^←(𝛼)=(𝑓,𝜑)^←(𝜑^′𝑜𝑝∘𝛼∘𝑓^′)=𝜑^𝑜𝑝∘𝜑^′𝑜𝑝∘𝛼∘𝑓^′∘𝑓=(𝜑^′∘𝜑)^𝑜𝑝∘𝛼∘𝑓^′∘𝑓=(𝑓^′∘𝑓,𝜑^′∘𝜑)^←(𝛼)=((𝑓^′,𝜑^′)∘(𝑓,𝜑))^←(𝛼).

Similar to the case of variety-based forward powerset operators, we have the following modifications of the functor introduced in Theorem 14.

Theorem 15. Suppose that either (Ξ₁,Ξ₂,†)=(⋁,⋀,⊢) or (Ξ₁,Ξ₂,†)=(⋀,⋁,⊣). Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(Ξ₁), every subcategory 𝐒 of 𝐋 provides the functor 𝐒𝐞𝐭×𝐒−−−−→^(−)†⇠(𝐂𝐒𝐋𝐚𝐭(Ξ₂))^𝑜𝑝, which is defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^†⇠=≀𝐿^𝑋1₁≀−−−−−−−−−→^{((𝑓,𝜑)†⇠)𝑜𝑝}≀𝐿^𝑋2₂≀, where (𝑓,𝜑)^†⇠(𝛼)=𝜑^†∘𝛼∘𝑓.

Proof. The proof is a straightforward modification of that of Theorem 14.

In the paper, we will use the following dualized versions of the functors of Theorems 12 and 13.

Theorem 16. (1)Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(⋁), every subcategory 𝐒 of 𝐋^𝑜𝑝 gives rise to the functor 𝐒𝐞𝐭^𝑜𝑝×𝐒−−−−−→^(−)→𝑜(𝐂𝐒𝐋𝐚𝐭(⋁))^𝑜𝑝 defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^→𝑜=‖𝐿^𝑋1₁‖−−−−−−−−−→^{((𝑓,𝜑)→𝑜)𝑜𝑝}‖𝐿^𝑋2₂‖, where ((𝑓,𝜑)^→𝑜(𝛼))(𝑥₁)=𝜑^𝑜𝑝(⋁_𝑓^𝑜𝑝_{(𝑥2)=𝑥1}𝛼(𝑥₂)). (2)Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(⋀), every subcategory 𝐒 of 𝐋 gives rise to the functor 𝐒𝐞𝐭^𝑜𝑝×𝐒−−−−−→^{(−)⊣⇢𝑜}(𝐂𝐒𝐋𝐚𝐭(⋁))^𝑜𝑝 defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^⊣⇢𝑜=≀𝐿^𝑋1₁≀−−−−−−−−−−→^{((𝑓,𝜑)⊣⇢𝑜)𝑜𝑝}≀𝐿^𝑋2₂≀, where ((𝑓,𝜑)^⊣⇢𝑜(𝛼))(𝑥₁)=𝜑^⊣(⋁_𝑓^𝑜𝑝_{(𝑥2)=𝑥1}𝛼(𝑥₂)).(3)Let 𝐋 be a variety, which extends 𝐂𝐒𝐋𝐚𝐭(⋁), and let 𝐒 be a subcategory of 𝐋 such that for every 𝐒-morphism 𝐿→−₁^𝜑𝐿₂, the map |𝐿₂|−−→^𝜑⊢|𝐿₁| is ⋁-preserving. Then there exists the functor 𝐒𝐞𝐭^𝑜𝑝×𝐒−−−−−→^{(−)⊢⇢𝑜}(𝐂𝐒𝐋𝐚𝐭(⋁))^𝑜𝑝 defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^⊢⇢𝑜=‖𝐿^𝑋1₁‖−−−−−−−−−−→^{((𝑓,𝜑)⊢⇢𝑜)𝑜𝑝}‖𝐿^𝑋2₂‖, where ((𝑓,𝜑)^⊢⇢𝑜(𝛼))(𝑥₁)=𝜑^⊢(⋁_𝑓^𝑜𝑝_{(𝑥2)=𝑥1}𝛼(𝑥₂)).

Additionally, the functors of Theorems 14 and 15 can be dualized as follows.

Theorem 17. (1) Given a variety 𝐀, every subcategory 𝐒 of 𝐀 induces the functor 𝐒𝐞𝐭^𝑜𝑝×𝐒−−−−→^(−)←𝑜𝐀, which is defined by ((𝑋₁,𝐴₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐴₂))^←𝑜=𝐴−−−−−−→^𝑋1₁^{(𝑓,𝜑)←𝑜}𝐴^𝑋2₂, where (𝑓,𝜑)^←𝑜(𝛼)=𝜑∘𝛼∘𝑓^𝑜𝑝.(2) Suppose that either (Ξ₁,Ξ₂,†)=(⋁,⋀,⊢) or (Ξ₁,Ξ₂,†)=(⋀,⋁,⊣). Given a variety 𝐋, which extends 𝐂𝐒𝐋𝐚𝐭(Ξ₁), every subcategory 𝐒 of 𝐋^𝑜𝑝 provides the functor 𝐒𝐞𝐭^𝑜𝑝×𝐒−−−−−→^{(−)†⇠𝑜}𝐂𝐒𝐋𝐚𝐭(Ξ₂), which is defined by ((𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂))^†⇠𝑜=≀𝐿^𝑋1₁≀−−−−−−−→^{(𝑓,𝜑)†⇠𝑜}≀𝐿^𝑋2₂≀, where (𝑓,𝜑)^†⇠𝑜(𝛼)=𝜑^†𝑜𝑝∘𝛼∘𝑓^𝑜𝑝.

The reader should notice that the functors of Theorem 17 are not standard in the fuzzy community.

2.3. Galois Connections

The paper relies heavily on the well-developed machinery of Galois connections. For convenience of the reader, we recall from [32, 37, 50] their basic definitions.

Definition 18. An order-preserving (resp., -reversing) Galois connection is a tuple ((𝑋₁,⩽),𝑓,𝑔,(𝑋₂,⩽)), where (𝑋₁,⩽), (𝑋₂,⩽) are posets and 𝑋⇄₁^𝑓_𝑔𝑋₂ are maps such that for every 𝑥₁∈𝑋₁ and every 𝑥₂∈𝑋₂, 𝑥₁⩽𝑔(𝑥₂) iff 𝑓(𝑥₁)⩽𝑥₂ (resp., 𝑥₁⩽𝑔(𝑥₂) iff 𝑥₂⩽𝑓(𝑥₁)).

As a consequence, one easily gets the following well-known features of Galois connections.

Theorem 19. Every order-preserving (resp., order-reversing) Galois connection ((𝑋₁,⩽),𝑓,𝑔,(𝑋₂,⩽)) has the following properties.(1) 1_𝑋1⩽𝑔∘𝑓 and 𝑓∘𝑔⩽1_𝑋2 (resp., 1_𝑋1⩽𝑔∘𝑓 and 1_𝑋2⩽𝑓∘𝑔). (2) Both 𝑓 and 𝑔 are order-preserving (resp., order-reversing). (3) 𝑓(⋁𝑆)=⋁_𝑠∈𝑆𝑓(𝑠) and 𝑔(⋀𝑇)=⋀_𝑡∈𝑇𝑔(𝑡) (resp., 𝑓(⋁𝑆)=⋀_𝑠∈𝑆𝑓(𝑠) and 𝑔(⋁𝑇)=⋀_𝑡∈𝑇𝑔(𝑡)) provided that the joins and meets in question exist in the domains. (4) 𝑔∘𝑓∘𝑔=𝑔 and 𝑓∘𝑔∘𝑓=𝑓.

Notice that the second item of Theorem 19 is responsible for the term “order-preserving’’ (resp., “order-reversing’’) in Definition 18. Moreover, in view of the results of Theorem 19, there exists the following equivalent definition of Galois connections.

Definition 20. An order-preserving (resp., order-reversing) Galois connection is provided by a tuple ((𝑋₁,⩽),𝑓,𝑔,(𝑋₂,⩽)), where (𝑋₁,⩽), (𝑋₂,⩽) are posets and (𝑋₁,⩽)⇄^𝑓_𝑔(𝑋₂,⩽) are order-preserving (resp., order-reversing) maps such that 1_𝑋1⩽𝑔∘𝑓 and 𝑓∘𝑔⩽1_𝑋2 (resp., 1_𝑋1⩽𝑔∘𝑓 and 1_𝑋2⩽𝑓∘𝑔).

To give the flavor of these notions, we show some examples of order-preserving Galois connections already encountered in the paper. An important example of order-reversing Galois connections will be encountered later on in the paper.

Example 21. (1) Every ⋁-semilattice (resp., ⋀-semilattice) homomorphism 𝐴→−₁^𝜑𝐴₂ provides the order-preserving Galois connection (𝐴₁,𝜑,𝜑^⊢,𝐴₂) (resp., (𝐴₂,𝜑^⊢,𝜑,𝐴₁)). (2) Every element 𝑞 of a quantale 𝑄 provides the order-preserving Galois connections (𝑄,⋅⊗𝑞,𝑞→_𝑙⋅,𝑄) and (𝑄,𝑞⊗⋅,𝑞→_𝑟⋅,𝑄). (3) Every map 𝑋−→₁^𝑓𝑋₂ provides the order-preserving Galois connection (𝒫(𝑋₁),𝑓^→,𝑓^←,𝒫(𝑋₂)). (4) Given a variety 𝐋 extending 𝐂𝐒𝐋𝐚𝐭(⋁), every 𝐒𝐞𝐭×𝐋-morphism (𝑋₁,𝐿₁)−−−−→^(𝑓,𝜑)(𝑋₂,𝐿₂) provides the order-preserving Galois connection (𝐿^𝑋1₁,(𝑓,𝜑)^→,(𝑓,𝜑)^⊢⇠,𝐿^𝑋2₂).

To conclude, we would like to notice that this paper studies the properties of the classical Galois connections on lattice-valued powersets. In contrast [38, 40], employ the notion of fuzzy Galois connection, which is particularly designed to fit the lattice-valued powerset framework. Such a modification is needed to preserve the main results valid on crisp powersets. The preservation in question comes, however, at the expense of a more complicated definition of fuzzy Galois connections themselves, which will be off our topic of study.

3. Variety-Based Topological Systems and Their Modifications

In [1], we introduced the concept of variety-based topological systems as a generalization of Vickers’ topological systems [2] mentioned in the Introduction section. Moreover, in [22], we presented a modified category of variety-based topological systems, to accommodate Vickers’ system localification procedure. For convenience of the reader, we recall both definitions, restating them, however, in a slightly more general way, to suit the framework of the current paper.

Definition 22. Given a variety 𝐀, a subcategory S of 𝐀^𝑜𝑝, and a reduct 𝐁 of 𝐀, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬 is the category, concrete over the product category 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒, which comprises the following data.

Objects. Tuples 𝐷=(pt𝐷,Ω𝐷,Σ𝐷,⊨) (𝐒_𝐁-topological systems or 𝐒_𝐁-systems), where (pt𝐷,Ω𝐷,Σ𝐷) is a 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒-object, whereas pt𝐷×Ω𝐷→−^⊨Σ𝐷 is a map (Σ𝐷-satisfaction relation on (pt𝐷,Ω𝐷)) such that Ω𝐷−−−−−→^{⊨(𝑥,−)}‖Σ𝐷‖ is a 𝐁-homomorphism for every 𝑥∈pt𝐷.

Morphisms. 𝐷−−−−−−−−−−−−−→₁^{𝑓=(pt𝑓,Ω𝑓,Σ𝑓)}𝐷₂ are all those 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒-morphisms (pt𝐷₁,Ω𝐷₁,Σ𝐷₁)−→^𝑓(pt𝐷₂, Ω𝐷₂,Σ𝐷₂), which satisfy the property ⊨₁(𝑥,(Ω𝑓)^𝑜𝑝(𝑏))=(Σ𝑓)^𝑜𝑝∘⊨₂(pt𝑓(𝑥),𝑏) for every 𝑥∈pt𝐷₁ and every 𝑏∈Ω𝐷₂ (𝐒_𝐁-continuity).

The following notational remark applies to all the categories of systems introduced in this paper.

Remark 23. The case 𝐒=𝐒_𝐴 is called (variety-based) fixed-basis approach, whereas all other cases are subsumed under (variety-based) variable-basis approach. Moreover, if 𝐀=𝐁, then the notation 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬 is truncated to 𝐒-𝐓𝐨𝐩𝐒𝐲𝐬, whereas if 𝐒=𝐒_𝐴, then the notation is shortened to 𝐴_𝐁-TopSys. In the latter case, the objects (resp., morphisms) are truncated to 𝐷=(pt𝐷,Ω𝐷,⊨) (resp., 𝑓=(pt𝑓,Ω𝑓)).

The next example gives the reader the intuition for the new category, showing that the latter incorporates many of the already existing concepts in the literature.

Example 24. (1) If 𝐀=𝐂𝐁𝐀𝐥𝐠, then the category 2_𝐅𝐫𝐦-𝐓𝐨𝐩𝐒𝐲𝐬, where 2={⟂,Τ} is the two-element complete Boolean algebra, is isomorphic to the category TopSys of topological systems of Vickers [2]. (2) If 𝐀=𝐁=𝐅𝐫𝐦, then the category 𝐋𝐨𝐜-𝐓𝐨𝐩𝐒𝐲𝐬 is the category of lattice-valued topological systems of Denniston et al. [8, 9]. (3) If 𝐀=𝐂𝐐𝐮𝐚𝐧𝐭, then the category 𝐴_𝐒𝐞𝐭-𝐓𝐨𝐩𝐒𝐲𝐬 is the category 𝐴-𝐈𝐧𝐭𝐒𝐲𝐬 of lattice-valued interchange systems of Denniston et al. [11, 29, 30], which is also the category 𝐴-𝐓𝐫𝐚𝐧𝐬𝐒𝐲𝐬 of lattice-valued transformer systems of Denniston et al. [15] (notice that both categories are fixed-basis). (4) If 𝐀=𝐁=𝐒𝐞𝐭, then the category 𝐴-𝐓𝐨𝐩𝐒𝐲𝐬 is the category 𝐂𝐡𝐮_𝐴 of Chu spaces over the set 𝐴 of Pratt [13]. In particular, the objects of 𝐂𝐡𝐮₂ are called formal contexts in Formal Concept Analysis of Ganter and Wille [31], whereas the objects of the category 𝐂𝐡𝐮_𝐴 itself are called many-valued formal contexts [31, Section 1.3]. Additionally, the respective context morphisms proposed by Ganter and Wille [31, Chapter 7] are different from the morphisms of Chu spaces, which will be dealt with in the subsequent sections of the paper.

Moreover, the following definition (extended and adapted for the current setting from [15]) illustrates the concept of a topological system morphism.

Definition 25. Letting 𝐀=𝐂𝐁𝐀𝐥𝐠, 𝐆𝐚𝐥𝐂𝐨𝐧 (notice that “GalCon’’ stands for “Galois connections’’) is the full subcategory of the category 2_𝐒𝐞𝐭-𝐓𝐨𝐩𝐒𝐲𝐬, whose objects are those 2_𝐒𝐞𝐭-topological systems 𝐷, which satisfy the following two conditions: (1)pt𝐷=Ω𝐷=𝑋; (2)⊨ is a partial order on 𝑋.

The reader can easily see (or verify) that the objects of the category 𝐆𝐚𝐥𝐂𝐨𝐧 are posets, and its morphisms are precisely the order-preserving Galois connections, which is reflected in its name. This category is essentially different from the category GAL of Galois connections of [32], in the sense that while the latter category uses Galois connections as its objects, the category 𝐆𝐚𝐥𝐂𝐨𝐧 uses them as morphisms. It will be the topic of our future research, to consider the properties of the category 𝐆𝐚𝐥𝐂𝐨𝐧 (at least) up to the extent of the respective ones of the category GAL. Notice, however, that the topic of the category GAL will find its proper place in the subsequent developments of the paper.

The reader should also be aware of the important fact that the case of order-reversing Galois connections (𝑋₁,⩽)⇄^𝑓_𝑔(𝑋₂,⩽) (up to the knowledge of the author) does not allow the previously mentioned system interpretation. Indeed, the already mentioned adjunction condition now reads as 𝑥₁⩽𝑔(𝑥₂) iff 𝑥₂⩽𝑓(𝑥₁) for every 𝑥₁∈𝑋₁ and every 𝑥₂∈𝑋₂. With the help of the dual partial order ⩽^𝑑 on 𝑋₂, we do get an order-preserving Galois connection (𝑋₁,⩽)⇄^𝑓_𝑔(𝑋₂,⩽^𝑑). It is unclear though how to incorporate in the system setting another order-reversing Galois connection (𝑋₂,⩽)⇄^𝑓′_𝑔′(𝑋₃,⩽) (or (𝑋₂,⩽)⇄^𝑓′_𝑔′(𝑋₃,⩽^𝑑)), that is, how to define the morphism composition law of the possible category (notice that (𝑋₂,⩽) and (𝑋₂,⩽^𝑑) are different as posets; i.e., the codomain of the first connection is not the domain of the second one).

In the next step, we introduce the modified category of topological systems inspired by [22].

Definition 26. Given a variety 𝐀, a subcategory 𝐒 of 𝐀, and a reduct 𝐁 of 𝐀, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑚 (notice that “𝑚’’ stands for “modified’’) is the category, concrete over the product category 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒, which comprises the following data.

Objects. Tuples 𝐷=(pt𝐷,Ω𝐷,Σ𝐷,⊨), where (pt𝐷,Ω𝐷,Σ𝐷) is a 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒-object and pt𝐷×Ω𝐷→−^⊨Σ𝐷 is a map such that Ω𝐷−−−−−→^{⊨(𝑥,−)}‖Σ𝐷‖ is a 𝐁-homomorphism for every 𝑥∈pt𝐷.

Morphisms. 𝐷−−−−−−−−−−−−−→₁^{𝑓=(pt𝑓,Ω𝑓,Σ𝑓)}𝐷₂ are 𝐒𝐞𝐭×𝐁^𝑜𝑝×𝐒-morphisms (pt𝐷₁,Ω𝐷₁,Σ𝐷₁)−→^𝑓(pt𝐷₂,Ω𝐷₂,Σ𝐷₂) such that Σ𝑓∘⊨₁(𝑥,(Ω𝑓)^𝑜𝑝(𝑏))=⊨₂(pt𝑓(𝑥),𝑏) for every 𝑥∈pt𝐷₁ and every 𝑏∈Ω𝐷₂.

Since both a category and its dual have the same objects, the only difference between the categories of systems introduced in Definitions 22 and 26 is in the direction of the Σ-component of their respective morphisms. In particular, with the notational conventions of Remark 23, we get that all the items of Example 24, except the second one, can be restated in terms of the category 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑚. Moreover, it is easy to provide the conditions, under which the system settings of Definitions 22 and 26 coincide.

Definition 27. Given a subcategory 𝐒 of 𝐀^𝑜𝑝, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬^𝑖 (resp., 𝐒_𝐁^𝑜𝑝-𝐓𝐨𝐩𝐒𝐲𝐬^𝑖_𝑚) is the nonfull subcategory of the category 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬 (resp., 𝐒_𝐁^𝑜𝑝-𝐓𝐨𝐩𝐒𝐲𝐬_𝑚), which has the same objects and whose morphisms 𝐷−→₁^𝑓𝐷₂ are such that Σ𝐷−−→₁^Σ𝑓Σ𝐷₂ (resp., Σ𝐷−−−−−→₁^{(Σ𝑓)𝑜𝑝}Σ𝐷₂) is an S- (resp., 𝐒^𝑜𝑝-) isomorphism.

Theorem 28. There exists the isomorphism 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬−→^𝑖𝐹𝐒_𝐁^𝑜𝑝-𝐓𝐨𝐩𝐒𝐲𝐬^𝑖_𝑚, which is defined by the formula 𝐹(𝐷−→₁^𝑓𝐷₂)=𝐷−−−−−−−−−−−−−−−−→₁^{(pt𝑓,Ω𝑓,((Σ𝑓)𝑜𝑝)−1)}𝐷₂.

Proof. Straightforward computations.

As a consequence, we obtain that there is no difference between the (fixed-basis) categories 𝐴_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬 and 𝐴_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑚. There exists, however, another obvious modification of the category of topological systems, which has never appeared in the literature before.

Definition 29. Given a variety 𝐀, a subcategory 𝐒 of 𝐀^𝑜𝑝, and a reduct 𝐁 of 𝐀, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎 (notice that “𝑎” stands for “alternative”) is the category, concrete over the product category 𝐒𝐞𝐭×𝐁×𝐒, which comprises the following data.
Objects. Tuples 𝐷=(pt𝐷,Ω𝐷,Σ𝐷,⊨), where (pt𝐷,Ω𝐷,Σ𝐷) is a 𝐒𝐞𝐭×𝐁×𝐒-object, whereas pt𝐷×Ω𝐷→−^⊨Σ𝐷 is a map such that Ω𝐷−−−−−→^{⊨(𝑥,−)}‖Σ𝐷‖ is a 𝐁-homomorphism for every 𝑥∈pt𝐷.
Morphisms. 𝐷−−−−−−−−−−−−−→₁^{𝑓=(pt𝑓,Ω𝑓,Σ𝑓)}𝐷₂ are 𝐒𝐞𝐭×𝐁×𝐒-morphisms  (pt𝐷₁,Ω𝐷₁,Σ𝐷₁)−→^𝑓(pt𝐷₂,Ω𝐷₂,Σ𝐷₂) such that ⊨₁(𝑥,𝑏)=(Σ𝑓)^𝑜𝑝∘⊨₂(pt𝑓(𝑥),Ω𝑓(𝑏)) for every 𝑥∈pt𝐷₁ and every 𝑏∈Ω𝐷₁.

It is important to emphasize that leaving (essentially) the same objects, we change the definition of morphisms dramatically. In particular, we cannot obtain a single item of Example 24 in the new framework.

Similar to the previously mentioned classical system setting, we can introduce a modified category of systems.

Definition 30. Given a variety 𝐀, a subcategory 𝐒 of 𝐀, and a reduct 𝐁 of 𝐀, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎𝑚 is the category, concrete over the product category 𝐒𝐞𝐭×𝐁×𝐒, which comprises the following data.
Objects. Tuples 𝐷=(pt𝐷,Ω𝐷,Σ𝐷,⊨), where (pt𝐷,Ω𝐷,Σ𝐷) is a 𝐒𝐞𝐭×𝐁×𝐒-object and pt𝐷×Ω𝐷→−^⊨Σ𝐷 is a map such that Ω𝐷−−−−−→^{⊨(𝑥,−)}‖Σ𝐷‖ is a 𝐁-homomorphism for every 𝑥∈pt𝐷.
Morphisms.𝐷−−−−−−−−−−−−−→₁^{𝑓=(pt𝑓,Ω𝑓,Σ𝑓)}𝐷₂ are 𝐒𝐞𝐭×𝐁×𝐒-morphisms (pt𝐷₁,Ω𝐷₁,Σ𝐷₁)−→^𝑓(pt𝐷₂,Ω𝐷₂,Σ𝐷₂) such that Σ𝑓∘⊨₁(𝑥,𝑏)=⊨₂(pt𝑓(𝑥),Ω𝑓(𝑏)) for every 𝑥∈pt𝐷₁ and every 𝑏∈Ω𝐷₁.

The explicit restatement of Theorem 28 for the setting of the categories 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎, 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎𝑚 is straightforward and is, therefore, left to the reader.

4. Lattice-Valued Formal Concept Analysis

For the required preliminaries in hand, in this section, we introduce different approaches to fuzzification of basic tools of Formal Concept Analysis (FCA) of Ganter and Wille [31]. In particular, we present three possible category-theoretic machineries, the first two of which stem from the approach of Denniston et al. [29], whereas the last one comes from Ganter and Wille themselves. For the sake of brevity, we omitted every motivational reason for the need of fuzziness in FCA, which is clearly stated in, for example, [33], and also could be found at the beginning of Section 5 of [29]. Our main interest is not the idea of FCA itself, but rather a better category-theoretic framework for its successful development.

4.1. Lattice-Valued Formal Contexts as Chu Spaces

In the last item of Example 24, we represented Chu spaces over sets of Pratt [13] as particular instances of variety-based topological systems. Moreover, we also mentioned that one of the main building blocks of FCA, that is, (many-valued) formal contexts, is, in fact, nothing else than Chu spaces (already noticed in the literature [55, 56]). With these ideas in mind, we introduce the following definition.

Definition 31. Let 𝐋 be an extension of the variety Quant of quantales and let 𝐁=𝐒𝐞𝐭. (1)𝐒-𝐅𝐂^𝐶 (notice that “𝐶” stands for “Chu”) is a new notation for the category 𝐒_𝐒𝐞𝐭-𝐓𝐨𝐩𝐒𝐲𝐬, whose objects ((lattice-valued) formal contexts) are now denoted 𝒦=(𝐺,𝑀,𝐿,𝐼) (in the notation of Definition 22, 𝒦=𝐷, which yields 𝐺=pt  𝐷,𝑀=Ω𝐷, 𝐿=Σ𝐷 and 𝐼=⊨), where 𝐺 (from German “Gegenstände”) is the set of context objects, 𝑀 (from German “Merkmale”) is the set of context attributes, and 𝐼 (from German “Inzidenzrelation”) is the context incidence relation. The respective morphisms ((lattice-valued) formal context morphisms) 𝒦−→₁^𝑓𝒦₂ are now triples (𝐺₁,𝑀₁,𝐿₁)−−−−−−−−→^{𝑓=(𝛼,𝛽,𝜑)}(𝐺₂,𝑀₂,𝐿₂) (in the notation of Definition 22, 𝛼=pt𝑓, 𝛽=Ω𝑓 and 𝜑=Σ𝑓, and then, for every 𝑔∈𝐺₁ and every 𝑚∈𝑀₂, 𝐼₁(𝑔,𝛽^𝑜𝑝(𝑚))=𝜑^𝑜𝑝∘𝐼₂(𝛼(𝑔),𝑚)). The case of 𝐋=𝐂𝐁𝐀𝐥𝐠 and 𝐒=𝐒₂ is called the crisp case. (2)𝐒-𝐅𝐂^𝐶_𝑚 is a new notation for the category 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑚.

The notational and naming remarks for objects and morphisms of Definition 31 will apply to all the categories of (lattice-valued) formal contexts introduced in this paper. Moreover, the reader can easily apply Theorem 28, to single out the isomorphic subcategories of the categories 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐶_𝑚.

At the moment, there is no agreement in the fuzzy community, whether to build formal contexts over L or 𝐋^𝑜𝑝, and, therefore, we just add the lower index “(−)_𝑚” in the former case. More precisely, this paper is (up to the knowledge of the author) the first attempt to develop a variable-basis approach (à la [4]) to lattice-valued FCA, the motivational article of Denniston et al. [29] being fixed-basis.

4.2. Lattice-Valued Formal Contexts in the Sense of Ganter and Wille

In the previous subsection, we presented a lattice-valued approach to FCA based in Chu spaces. An experienced reader, however, may be well aware of many-valued contexts of Ganter and Wille themselves [31, Section  13], the theory of which is already established in the literature. In the following, we provide its category-theoretic analogue, bringing together both many-valued contexts and their morphisms.

Definition 32. Let 𝐋 be an extension of 𝐐𝐮𝐚𝐧𝐭 and let 𝐁=𝐒𝐞𝐭. (1)𝐒-𝐅𝐂^𝐺𝑊(notice that “𝐺𝑊” stands for “Ganter and Wille”) is a new notation for the category 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎. (2)𝐒-𝐅𝐂_𝑚^𝐺𝑊 is a new notation for the category 𝐒_𝐁-𝐓𝐨𝐩𝐒𝐲𝐬_𝑎𝑚.

It should be emphasized immediately that neither Ganter nor Wille (up to the knowledge of the author) has considered a categorical approach to (many-valued) formal contexts, that is, has united both formal contexts and their morphisms in one entity. Additionally, both (many-valued) formal contexts and their morphisms are treated in an algebraic way in [31] (from where our notation for context morphisms, which is different from the system setting, is partly borrowed). There does exist some research on certain categories of contexts (e.g., [57–61]), which, however, does not study the categorical foundation for FCA, but rather concentrates on a fixed categorical framework and its related results. It is the main purpose of this paper, to investigate such possible category-theoretic foundations and their relationships to each other (leaving their related results to the further study on the topic). In particular, we consider a category of contexts, whose morphisms are strikingly different from the standard framework of topological systems.

4.3. Lattice-Valued Formal Contexts in the Sense of Denniston et al.

In [29], Denniston et al. have introduced another approach to lattice-valued FCA, which was based on the category GAL of order-preserving Galois connections of [32]. In the following, we extend their approach in two respects. Firstly, we employ the variety Quant of quantales instead of CQuant of commutative ones of [29] (called there complete commutative residuated lattices). Secondly, we provide a variable-basis approach (à la [4]) to the topic, instead of the fixed-based one of [29].

To begin with, we introduce some new notions and notations (induced by the respective ones of [29]), which will be used in the subsequent developments of the paper.

Definition 33. Every lattice-valued formal context 𝒦 has the following (lattice-valued) Birkhoff operators: (1)𝐿−−→^𝐺𝐻𝐿^𝑀, which is given by (𝐻(𝑠))(𝑚)=⋀_𝑔∈𝐺(𝑠(𝑔)→_𝑙𝐼(𝑔,𝑚)),(2)𝐿−→^𝑀𝐾𝐿^𝐺, which is given by (𝐾(𝑡))(𝑔)=⋀_𝑚∈𝑀(𝑡(𝑚)→_𝑟𝐼(𝑔,𝑚)).

Given a lattice-valued context 𝒦, the notation 𝑠 for the elements of 𝐿^𝐺 (resp., 𝑡 for the elements of 𝐿^𝑀), that is, lattice-valued sets of context objects (resp., attributes), will be used throughout the paper.

To give the reader more intuition, we provide an example of the just introduced notions.

Example 34. Every crisp context 𝒦 (recall our crispness remark from Definition 31(1)) provides the maps (1)𝒫(𝐺)−−→^𝐻𝒫(𝑀), which is given by 𝐻(𝑆)={𝑚∈𝑀∣𝑠𝐼𝑚forevery𝑠∈𝑆},(2)𝒫(𝑀)−→^𝐾𝒫(𝐺), which is given by 𝐾(𝑇)={𝑔∈𝐺∣𝑔𝐼𝑡forevery𝑡∈𝑇}, which are the classical Birkhoff operators generated by a binary relation.

An important property of the maps of Definition 33 is contained in the following result.

Theorem 35. For every lattice-valued context 𝒦, (𝐿^𝐺,𝐻,𝐾,𝐿^𝑀) is an order-reversing Galois connection.

Proof. For convenience of the reader, we provide the proof, which is different from that of [29].
To show that the map 𝐻 is order-reversing, notice that given 𝑠₁, 𝑠₂∈𝐿^𝐺 such that 𝑠₁⩽𝑠₂, and some 𝑚∈𝑀, for every 𝑔∈𝐺, it follows that 𝑠₂(𝑔)→_𝑙𝐼(𝑔,𝑚)⩽𝑠₁(𝑔)→_𝑙𝐼(𝑔,𝑚) (Theorem  6(2))  and,  therefore,  (𝐻(𝑠₂))(𝑚)=⋀_𝑔∈𝐺(𝑠₂(𝑔)→_𝑙𝐼(𝑔,𝑚))⩽⋀_𝑔∈𝐺(𝑠₁(𝑔)→_𝑙𝐼(𝑔,𝑚))=(𝐻(𝑠₁))(𝑚). The case of 𝐾 is similar.
To show that 1_𝐿^𝐺⩽𝐾∘𝐻, notice that given 𝑠∈𝐿^𝐺 and 𝑔∈𝐺, it follows that (𝐾∘𝐻(𝑠))(𝑔)=⋀_𝑚∈𝑀((𝐻(𝑠))(𝑚)→_𝑟𝐼(𝑔,𝑚))=⋀_𝑚∈𝑀((⋀_{𝑔′∈𝐺}(𝑠(𝑔^′)→_𝑙𝐼(𝑔^′,𝑚)))→_𝑟⋀𝐼(𝑔,𝑚))⩾_𝑚∈𝑀((𝑠(𝑔)→_𝑙𝐼(𝑔,𝑚))→_𝑟𝐼(𝑔,𝑚))⩾^(†)⋀_𝑚∈𝑀𝑠(𝑔)⩾𝑠(𝑔) , where (†) relies on the first claim of Theorem 6(3). The proof of 1_𝐿^𝑀⩽𝐻∘𝐾 uses the dual machinery and the second claim of Theorem 6(3).

We emphasize immediately that Theorem 35 is an analogue of the already obtained results of [38, Lemma  3] (for 𝐿 a complete residuated lattice) and of [40, Proposition  3.9] (for 𝐿 a unital quantale) in the framework of fuzzy Galois connections, which is different (more demanding) from the setting of Galois connections of the current paper (see additionally the remark after Theorem 42). There also exists an extension of Theorem 35 to a more general context of (commutative, right-distributive, complete) extended-order algebras 𝐿 (whose multiplication in some cases need not be associative) [62, Propositions  3.9, 5.5] and, additionally, to even more general case, which involves relations instead of maps (the so-called 𝐿-Galois triangles) [63].

Everything is now in hand, to define another lattice-valued extension of formal contexts of FCA.

Definition 36. Given a variety L, which extends Quant, and a subcategory S of L, 𝐒-𝐅𝐂^𝐷𝑀𝑅 (notice that “DMR” stands for “Denniston, Melton and Rodabaugh”) is the category, concrete over the product category 𝐒𝐞𝐭×𝐒𝐞𝐭^𝑜𝑝, which comprises the following data.
Objects. Lattice-valued formal contexts 𝒦 (in the sense of Definition 31) such that 𝐿 is an object of S.
Morphisms. 𝒦−−−−−−→₁^{𝑓=(𝛼,𝛽)}𝒦₂ are 𝐒𝐞𝐭×𝐒𝐞𝐭^𝑜𝑝-morphisms (𝐿^𝐺1₁,𝐿^𝑀1₁)−−−−→^(𝛼,𝛽)(𝐿^𝐺2₂,𝐿^𝑀2₂), which make the diagrams (4) commute.

It should be emphasized that Definition 36 is inspired by the respective one of [29], which, in its turn, comes from the definition of the category GAL of order-preserving Galois connections of [32]. Moreover, two important features of the category 𝐒-𝐅𝐂^𝐷𝑀𝑅 are worth mentioning. Firstly, it never depends on the morphisms of the category 𝐒, mentioned at the beginning of its definition (𝐒-objects, however, are employed), which is reflected well enough in its truncated ground category. Secondly, its underlying functor to the ground category 𝐒𝐞𝐭×𝐒𝐞𝐭^𝑜𝑝 is defined on objects as |𝒦|=(𝐿^𝐺,𝐿^𝑀), which is a huge difference from the already introduced categories of formal contexts (based on the machinery of topological systems), the underlying functor of which is just |𝒦|=(𝐺,𝑀,𝐿).

By analogy with the already presented categories of lattice-valued formal contexts, we introduce another version of the category of Definition 36. Unlike the case of the categories 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐺𝑊, where the change concerns the direction of the algebraic part of their morphisms, this time we change the set-theoretic morphism part and, therefore, use the notation “(−)_𝑎” (“alternative”) instead of “(−)_𝑚” (“modified”).

Definition 37. Given a variety 𝐋, which extends Quant, and a subcategory 𝐒 of 𝐋, 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 is the category, concrete over the product category 𝐒𝐞𝐭×𝐒𝐞𝐭, which comprises the following data.
Objects. Lattice-valued formal contexts 𝒦 (in the sense of Definition 31) such that 𝐿 is an object of 𝐒.
Morphisms. 𝒦−−−−−−→₁^{𝑓=(𝛼,𝛽)}𝒦₂ are 𝐒𝐞𝐭×𝐒𝐞𝐭-morphisms (𝐿^𝐺1₁,𝐿^𝑀1₁)−−−−→^(𝛼,𝛽)(𝐿^𝐺2₂,𝐿^𝑀2₂), which make the diagrams (5) commute.

In the following, we adapt the technique of Theorem 28, to single out the possible isomorphic subcategories of the categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅.

Definition 38.  𝐒-𝐅𝐂_𝑖^𝐷𝑀𝑅 (resp., 𝐒-𝐅𝐂^𝐷𝑀𝑅_𝑎𝑖) is the nonfull subcategory of the category 𝐒-𝐅𝐂^𝐷𝑀𝑅 (resp., 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅), which has the same objects and whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that 𝐿−−−→^𝑀2₂^𝛽𝑜𝑝𝐿^𝑀1₁ (resp., 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂) is an isomorphism in 𝐒𝐞𝐭, that is, bijective.

Theorem 39. There exists the isomorphism 𝐒-𝐅𝐂_𝑖^{𝐹𝐷𝑀𝑅}−→𝐒-𝐅𝐂^𝐷𝑀𝑅_𝑎𝑖, which is defined by the formula 𝐹(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−→₁^{(𝛼,(𝛽𝑜𝑝)−1)}𝒦₂.

The reader should be aware that the fixed-basis approach (i.e., the case of 𝐒=𝐒_𝐿) will (in general) provide nonisomorphic categories 𝐿-𝐅𝐂^𝐷𝑀𝑅 and 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅, since both of them are not dependant on S-morphisms. More precisely, it is possible to obtain the following result.

Theorem 40. Suppose that 𝐿 is not a singleton, and, additionally, its quantale multiplication is not a constant map to ⟂_𝐿. Then the categories 𝐿-𝐅𝐂^𝐷𝑀𝑅 and 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅 are nonisomorphic.

Proof. It is easy to see that the context 𝒦=(𝑋,𝑋,𝐼), where 𝑋 is the empty set and 𝑋×𝑋→−^𝐼𝐿 is the unique possible map, provides a terminal object in the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅. The category 𝐿-𝐅𝐂^𝐷𝑀𝑅, however, does not have a terminal object, which can be shown as follows.
Suppose the context 𝒦_𝑇=(𝐺_𝑇,𝑀_𝑇,𝐼_𝑇) is terminal in 𝐿-𝐅𝐂^𝐷𝑀𝑅. Define a context 𝒦^′=(𝑋^′,𝑋^′,𝐼^′), where 𝑋^′ is a nonempty set and 𝐼^′=⊤_𝐿 (the constant map with value ⊤_𝐿). One gets two equal constant maps 𝐿−−−−−−−−−→^{𝑋′𝐻′=𝐾′=⊤𝐿}𝐿^𝑋′. Since there exists an 𝐿-𝐅𝐂^𝐷𝑀𝑅-morphism 𝒦−−−−−−→^{′𝑓=(𝛼,𝛽)}𝒦_𝑇, the diagrams (6) commute. It follows that the map 𝐾_𝑇 in the right-hand diagram is constant (since 𝐾^′ has that property). Thus, if 𝐿−−−→^{𝑀𝑇𝛽′𝑜𝑝}𝐿^𝑋′ maps everything to ⊤_𝐿, then every map 𝐿−→^{𝑋′𝛼′}𝐿^𝐺𝑇, taking ⊤_𝐿 to the constant value of the map 𝐾_𝑇, gives another 𝐿-𝐅𝐂^𝐷𝑀𝑅-morphism 𝒦−−→^′𝑓′𝒦_𝑇. The uniqueness of such morphism necessitates 𝐿^𝐺𝑇 to be a singleton, and then 𝐺_𝑇 is the empty set (since 𝐿 is not a singleton). As a consequence, we get that the map 𝐻_𝑇 is constant as well (with the value ⊤_𝐿). Define now another context 𝒦^′′=(𝑋^′′,𝑋^′′,𝐼^′′), where 𝑋^′′ is a nonempty set and 𝐼^′′=⟂_𝐿. Since the quantale multiplication of 𝐿 is not the constant map to ⟂_𝐿, we get a nonconstant map 𝐻^′′ (𝐻^′′(⟂_𝐿)=⊤_𝐿≠𝐻^′′(⊤_𝐿)), and, therefore, a contradiction, since the left-hand side of the previous two diagrams factorizes then (with the help of some 𝐿-𝐅𝐂^𝐷𝑀𝑅-morphism 𝒦−−→^{𝑓′′′′}𝒦_𝑇) the nonconstant map 𝐻^′′ through the constant map 𝐻_𝑇.

The case when 𝐿 is a singleton clearly provides the isomorphic categories 𝐿-𝐅𝐂^𝐷𝑀𝑅 and 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅. The case when 𝐿 is equipped with the constant ⟂_𝐿-valued quantale multiplication results in all context maps 𝐻, 𝐾 being constant (with the value ⊤_𝐿, which should additionally be preserved by the corresponding maps 𝛼 and 𝛽^𝑜𝑝 (resp., 𝛽)), which does not fit the machinery of Theorem 40, and will be treated in our subsequent articles on lattice-valued FCA. We should emphasize, however, immediately that the just mentioned two cases are not that interesting (i.e., are usually skipped by the researchers), and, therefore, we can rightly conclude that the approaches of Definitions 36 and 37 are fundamentally different, which provides a (partial) justification for our alternative setting of the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅.

The difference between the categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 is similar to that between the categories 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐺𝑊 (which served as our main motivation for introducing the approach of 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which was mentioned, but never studied, in [29]).

4.4. Some Properties of Lattice-Valued Birkhoff Operators

It is a well-known (and easy to establish) fact that all order-reversing Galois connections between crisp powersets arise from a pair of crisp Birkhoff operators of Example 34. In the following, we show that the lattice-valued case destroys partly this essential property (the reader is advised to recall the notational remarks concerning particular maps of the powerset algebras, mentioned at the beginning of Section 2.2).

Theorem 41. If 𝐋 extends 𝐔𝐐𝐮𝐚𝐧𝐭, then every lattice-valued context 𝒦 satisfies the following. (1)(𝐻(𝜒^1𝐿_{𝑔}))(𝑚)=(𝐾(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺 and every 𝑚∈𝑀. (2)(𝐻(𝑎⊗𝜒^1𝐿_{𝑔}))(𝑚)=𝑎→_𝑙(𝐻(𝜒^1𝐿_{𝑔}))(𝑚) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿. (3)(𝐾(𝜒^1𝐿_{𝑚}⊗𝑎))(𝑔)=𝑎→_𝑟(𝐾(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿.
The previously mentioned items are equivalent to the following ones. (a)(𝐻(𝑎⊗𝜒^1𝐿_{𝑔}))(𝑚)=𝑎→_𝑙(𝐾(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿. (b)(𝐾(𝜒^1𝐿_{𝑚}⊗𝑎))(𝑔)=𝑎→_𝑟(𝐻(𝜒^1𝐿_{𝑔}))(𝑚) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿.

Proof. Ad (1). The item follows from the fact that (𝐻(𝜒^1𝐿_{𝑔}))(𝑚)=⋀_{𝑔′∈𝐺}(𝜒^1𝐿_{𝑔}(𝑔^′)→_𝑙𝐼(𝑔^′,𝑚))=^(†)1_𝐿→_𝑙𝐼(𝑔,𝑚)=𝐼(𝑔,𝑚)=1_𝐿→_𝑟𝐼(𝑔,𝑚)=^(†)⋀_{𝑚′∈𝑀}(𝜒^1𝐿_{𝑚}(𝑚^′)→_𝑟𝐼(𝑔,𝑚^′))=(𝐾(𝜒^1𝐿_{𝑚}))(𝑔), where (†) uses Theorem 6(2).
Ad (2). (𝐻(𝑎⊗𝜒^1𝐿_{𝑔}))(𝑚)=⋀_{𝑔′∈𝐺}((𝑎⊗𝜒^1𝐿_{𝑔})(𝑔^′)→_𝑙𝐼(𝑔^′,𝑚))=⋀_{𝑔′∈𝐺}((𝑎⊗𝜒^1𝐿_{𝑔}(𝑔^′))→_𝑙𝐼(𝑔^′,𝑚))=^(†)⋀_{𝑔′∈𝐺}(𝑎→_𝑙(𝜒^1𝐿_{𝑔}(𝑔^′)→_𝑙𝐼(𝑔^′,𝑚)))=^(††)𝑎→_𝑙(⋀_{𝑔′∈𝐺}(𝜒^1𝐿_{𝑔}(𝑔^′)→_𝑙𝐼(𝑔^′,𝑚)))=𝑎→_𝑙(𝐻(𝜒^1𝐿_{𝑔}))(𝑚), where (†) (resp., (††)) uses item (4) (resp., item (1)) of Theorem 6.
Ad (3). (𝐾(𝜒^1𝐿_{𝑚}⊗𝑎))(𝑔)=⋀_{𝑚′∈𝑀}((𝜒^1𝐿_{𝑚}⊗𝑎)(𝑚^′)→_𝑟𝐼(𝑔,𝑚^′))=⋀_{𝑚′∈𝑀}((𝜒^1𝐿_{𝑚}(𝑚^′)⊗𝑎)→_𝑟𝐼(𝑔,𝑚^′))=^(†)⋀_{𝑚′∈𝑀}(𝑎→_𝑟(𝜒^1𝐿_{𝑚}(𝑚^′)→_𝑟𝐼(𝑔,𝑚^′)))=^(††)𝑎→_𝑟(⋀_{𝑚′∈𝑀}(𝜒^1𝐿_{𝑚}(𝑚^′)→_𝑟𝐼(𝑔,𝑚^′)))=𝑎→_𝑟(𝐾(𝜒^1𝐿_{𝑚}))(𝑔), where (†) (resp., (††)) uses item (4) (resp., item (1)) of Theorem 6.
Ad (1),(2),(3)⇒(𝑎),(𝑏). Use (1) in the right-hand side of both (2) and (3).
Ad (𝑎),(𝑏)⇒(1),(2),(3). Put 𝑎=1_𝐿 in any of (𝑎) or (𝑏) and get (1). Now, use (1) in the right-hand side of both (𝑎) and (𝑏), to get (2) and (3), respectively.

Theorem 41 paves the way for a partial generalization of the previously mentioned result on generating order-reversing Galois connections between powersets.

Theorem 42. Let 𝐺, 𝑀 be sets and let 𝐿 be a unital quantale. For every order-reversing Galois connection (𝐿^𝐺,𝛼,𝛽,𝐿^𝑀), the following statements are equivalent.(1) There exists a map 𝐺×𝑀→−^𝐼𝐿 such that 𝛼=𝐻 and 𝛽=𝐾.(2)(a) (𝛼(𝜒^1𝐿_{𝑔}))(𝑚)=(𝛽(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺 and every 𝑚∈𝑀. (b)(𝛼(𝑎⊗𝜒^1𝐿_{𝑔}))(𝑚)=𝑎→_𝑙(𝛼(𝜒^1𝐿_{𝑔}))(𝑚) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿. (c)(𝛽(𝜒^1𝐿_{𝑚}⊗𝑎))(𝑔)=𝑎→_𝑟(𝛽(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿.(3)(a) (𝛼(𝑎⊗𝜒^1𝐿_{𝑔}))(𝑚)=𝑎→_𝑙(𝛽(𝜒^1𝐿_{𝑚}))(𝑔) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿. (b) (𝛽(𝜒^1𝐿_{𝑚}⊗𝑎))(𝑔)=𝑎→_𝑟(𝛼(𝜒^1𝐿_{𝑔}))(𝑚) for every 𝑔∈𝐺, 𝑚∈𝑀 and 𝑎∈𝐿.

Proof. (1)⇒(2) and (2)⇔(3). It follows from Theorem 41.
(2)⇒(1). Define the required map 𝐺×𝑀→−^𝐼𝐿 by 𝐼(𝑔,𝑚)=(𝛼(𝜒^1𝐿_{𝑔}))(𝑚)=^(𝑎)(𝛽(𝜒^1𝐿_{𝑚}))(𝑔). Given 𝑠∈𝐿^𝐺 and 𝑚∈𝑀, it follows that (𝐻(𝑠))(𝑚)=⋀_𝑔∈𝐺(𝑠(𝑔)→_𝑙𝐼(𝑔,𝑚))=⋀_𝑔∈𝐺(𝑠(𝑔)→_𝑙(𝛼(𝜒^1𝐿_{𝑔}))(𝑚))=^(𝑏)⋀_𝑔∈𝐺(𝛼(𝑠(𝑔)⊗𝜒^1𝐿_{𝑔}))(𝑚)=^(†)(𝛼(⋁_𝑔∈𝐺(𝑠(𝑔)⊗𝜒^1𝐿_{𝑔})))(𝑚)=(𝛼(𝑠))(𝑚) (notice that (⋁_𝑔∈𝐺(𝑠(𝑔)⊗𝜒^1𝐿_{𝑔}))(𝑔^′)=⋁_𝑔∈𝐺(𝑠(𝑔)(𝑔^′)⊗𝜒^1𝐿_{𝑔}(𝑔^′))=𝑠(𝑔^′)⊗1_𝐿=𝑠(𝑔^′) for every 𝑔^′∈𝐺), whereas, given 𝑡∈𝐿^𝑀 and 𝑔∈𝐺, it follows that (𝐾(𝑡))(𝑔)=⋀_𝑚∈𝑀(𝑡(𝑚)→_𝑟𝐼(𝑔,𝑚))=⋀_𝑚∈𝑀(𝑡(𝑚)→_𝑟(𝛽(𝜒^1𝐿_{𝑚}))(𝑔))=^(𝑐)⋀_𝑚∈𝑀(𝛽(𝜒^1𝐿_{𝑚}⊗𝑡(𝑚)))(𝑔)=^(†)(𝛽(⋁_𝑚∈𝑀(𝜒^1𝐿_{𝑚}⊗𝑡(𝑚))))(𝑔)=(𝛽(𝑡))(𝑔), where (†) uses Theorem 19(3).

Theorems 41 and 42 show that in the lattice-valued setting, every map 𝐺×𝑀→−^𝐼𝐿 provides an order-reversing Galois connection, whereas the converse way is possible under certain restrictions only. As a simple counterexample, consider the case, when 𝐿 is the unit interval 𝕀=([0,1],⋁,⋀,1), and both 𝐺 and 𝑀 are singletons. We can, therefore, assume that both 𝕀^𝐺 and 𝕀^𝑀 are the same as 𝕀. The order-reversing involution map 𝕀→−^𝛼𝕀, 𝛼(𝑎)=1−𝑎 is a part of the order-reversing Galois connection (𝕀,𝛼,𝛼,𝕀). It is easy to see that the condition of, for example, Theorem 42(3)(a) amounts to 𝛼(𝑎)=𝑎→𝛼(1) for every 𝑎∈𝕀 (notice that the quantale in question is commutative). However, for 𝑎=1/2, we obtain that 𝛼(1/2)=1/2≠0=1/2→0=1/2→𝛼(1).

We also would like to emphasize here that [40] uses a different approach to Galois connections (called polarities of certain types), which has the advantage of still preserving the one-to-one correspondence between them and Birkhoff operators even in the lattice-valued case [40, Theorem  3.11]. The underlying framework of such polarities, however, is more demanding than that of this paper (e.g., requires enriched categories) and, therefore, will be not considered here. Also notice that [64, Proposition  7.1] provides an analogue of Theorem 42 for the case of strictly two-sided commutative quantales, considering, however, lattice-valued versions of the powerset operators generated by a relation and extending those given at the beginning of Section 2.2 and, therefore, dealing with order-preserving Galois connections. More precisely, in the notation of [64], every formal context 𝒦=(𝐺,𝑀,𝐿,𝐼) (where 𝐿 is additionally assumed to be a strictly two-sided commutative quantale) provides the following generalizations of the standard powerset operators: (1)𝐿−−→^𝐺𝐼⊗𝐿^𝑀 given by (𝐼^⊗(𝑠))(𝑚)=⋁_𝑔∈𝐺𝐼(𝑔,𝑚)⊗𝑠(𝑔) (forward powerset operator),(2)𝐿−−−^{𝑀𝐼→−}→𝐿^𝐺 given by (𝐼^→₋(𝑡))(𝑔)=⋀_𝑚∈𝑀𝐼(𝑔,𝑚)→𝑡(𝑚) (backward powerset operator). In particular, (𝐿^𝐺,𝐼^⊗,𝐼^→₋,𝐿^𝑀) is an order-preserving Galois connection (recall Definition 18). It is the main purpose of [64, Proposition  7.1] to clarify the conditions, when an order-preserving Galois connection (𝐿^𝐺,𝛼,𝛽,𝐿^𝑀) has the previous form, that is, is generated by a map 𝐺×𝑀→−^𝐼𝐿.

We end the subsection with another simple (but useful) property of lattice-valued Birkhoff operators.

Lemma 43. Given a lattice-valued formal context 𝒦 such that 𝐼=⊤_𝐿, it follows that 𝐻=⊤_𝐿 and 𝐾=⊤_𝐿. In particular, in the crisp case, 𝐻=𝑀 and 𝐾=𝐺.

Proof. It is enough to consider the case of 𝐻 (that of 𝐾 being similar). Given 𝑠∈𝐿^𝐺, for every 𝑚∈𝑀, (𝐻(𝑠))(𝑚)=⋀_𝑔∈𝐺(𝑠(𝑔)→_𝑙𝐼(𝑔,𝑚))=⋀_𝑔∈𝐺(𝑠(𝑔)→_𝑙⊤_𝐿)=⋀_𝑔∈𝐺⊤_𝐿=⊤_𝐿, that is, 𝐻(𝑠)=⊤_𝐿.

4.5. Lattice-Valued Formal Concepts, Protoconcepts, and Preconcepts

In the developments of [29], Denniston et al. use lattice-valued extensions of the notions of formal concept, protoconcept, and preconcept of, for example, [36]. In the following, we recall their respective definitions and show some of their properties in our more general framework, which will be used later on in the paper.

Definition 44. Let 𝒦 be a lattice-valued formal context and let 𝑠∈𝐿^𝐺, 𝑡∈𝐿^𝑀. The pair (𝑠,𝑡) is called a (1)(lattice-valued) formal concept of 𝒦 provided that 𝐻(𝑠)=𝑡 and 𝐾(𝑡)=𝑠,(2)(lattice-valued) formal protoconcept of 𝒦 provided that 𝐾∘𝐻(𝑠)=𝐾(𝑡) (equivalently, 𝐻∘𝐾(𝑡)=𝐻(𝑠)),(3)(lattice-valued) formal preconcept of 𝒦 provided that 𝑠⩽𝐾(𝑡) (equivalently, 𝑡⩽𝐻(𝑠)).

Using the properties of Galois connections, mentioned in Theorem 19, one can easily get that every concept is a protoconcept, and every protoconcept is a preconcept, whereas the converse implications, in general, are not true. Moreover, the next result provides simple properties of the notions of Definition 44 with respect to the morphisms of the previously mentioned categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅.

Theorem 45. Let 𝒦−→₁^𝑓𝒦₂ be a morphism of the category 𝐒-𝐅𝐂^𝐷𝑀𝑅 and let 𝑠₁∈𝐿^𝐺1₁, 𝑡₂∈𝐿^𝑀2₂. (1)(𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal concept of 𝒦₁ and 𝐻₂∘𝐾₂(𝑡₂)=𝑡₂ iff (𝛼(𝑠₁),𝑡₂) is a formal concept of 𝒦₂ and 𝐾₁∘𝐻₁(𝑠₁)=𝑠₁. (2)If both 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ and 𝐿−−−→^𝑀2₂^𝛽𝑜𝑝𝐿^𝑀1₁ reflect fixed points, that is, 𝐾₂∘𝐻₂∘𝛼(𝑠)=𝛼(𝑠) implies 𝐾₁∘𝐻₁(𝑠)=𝑠 for every 𝑠∈𝐿^𝐺1₁, and 𝐻₁∘𝐾₁∘𝛽^𝑜𝑝(𝑡)=𝛽^𝑜𝑝(𝑡) implies 𝐻₂∘𝐾₂(𝑡)=𝑡 for every 𝑡∈𝐿^𝑀2₂, then (𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal concept of 𝒦₂. (3)If both 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ and 𝐿−−−→^𝑀2₂^𝛽𝑜𝑝𝐿^𝑀1₁ are injective, then (𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal concept of 𝒦₂. (4)(𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal protoconcept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal protoconcept of 𝒦₂. (5)If both 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ and 𝐿−−−→^𝑀2₂^𝛽𝑜𝑝𝐿^𝑀1₁ are order-preserving, then (𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal preconcept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal preconcept of 𝒦₂.
If 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ (resp., 𝐿−−−→^𝑀2₂^𝛽𝑜𝑝𝐿^𝑀1₁) is injective, then 𝛼 (resp., 𝛽) reflects fixed points in the sense of (2).

Proof. Ad (1). “⇒”: Since both 𝑠₁=𝐾₁∘𝛽^𝑜𝑝(𝑡₂) and 𝐻₁(𝑠₁)=𝛽^𝑜𝑝(𝑡₂), we get 𝛼(𝑠₁)=𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐾₂(𝑡₂), 𝐻₂∘𝛼(𝑠₁)=𝐻₂∘𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐻₂∘𝐾₂(𝑡₂)=𝑡₂, and, moreover, 𝐾₁∘𝐻₁(𝑠₁)=𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝑠₁. “⇐’’: Since both 𝐻₂∘𝛼(𝑠₁)=𝑡₂ and 𝛼(𝑠₁)=𝐾₂(𝑡₂), we get 𝛽^𝑜𝑝(𝑡₂)=𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠₁)=𝐻₁(𝑠₁), 𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐾₁∘𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠₁)=𝐾₁∘𝐻₁(𝑠₁)=𝑠₁, and, moreover, 𝐻₂∘𝐾₂(𝑡₂)=𝐻₂∘𝛼(𝑠₁)=𝑡₂.
Ad (2). “⇒”: 𝛽^𝑜𝑝(𝑡₂)=𝐻₁(𝑠₁) yields 𝐻₁∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐻₁∘𝐾₁∘𝐻₁(𝑠₁)=𝐻₁(𝑠₁)=𝛽^𝑜𝑝(𝑡₂), and, therefore, 𝐻₂∘𝐾₂(𝑡₂)=𝑡₂, since 𝛽^𝑜𝑝 reflects fixed points. Now, employ item (1). “⇐”: 𝐾₂(𝑡₂)=𝛼(𝑠₁) yields 𝐾₂∘𝐻₂∘𝛼(𝑠₁)=𝐾₂∘𝐻₂∘𝐾₂(𝑡₂)=𝐾₂(𝑡₂)=𝛼(𝑠₁), and, therefore, 𝐾₁∘𝐻₁(𝑠₁)=𝑠₁, since 𝛼 reflects fixed points. Now, employ item (1).
Ad (3). It follows from (2) and the last claim of the theorem.
Ad (4). “⇒”: Since 𝐾₁∘𝐻₁(𝑠₁)=𝐾₁∘𝛽^𝑜𝑝(𝑡₂), we get 𝐾₂∘𝐻₂∘𝛼(𝑠₁)=𝛼∘𝐾₁∘𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠₁)=𝛼∘𝐾₁∘𝐻₁(𝑠₁)=𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐾₂(𝑡₂). “⇐”: Since 𝐻₂∘𝐾₂(𝑡₂)=𝐻₂∘𝛼(𝑠₁), we get 𝐻₁∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝛽^𝑜𝑝∘𝐻₂∘𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝛽^𝑜𝑝∘𝐻₂∘𝐾₂(𝑡₂)=𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠₁)=𝐻₁(𝑠₁).
Ad (5). “⇒”: 𝑠₁⩽𝐾₁∘𝛽^𝑜𝑝(𝑡₂) implies 𝛼(𝑠₁)⩽𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡₂)=𝐾₂(𝑡₂) (𝛼 is order-preserving). “⇐”: 𝑡₂⩽𝐻₂∘𝛼(𝑠₁) yields 𝛽^𝑜𝑝(𝑡₂)⩽𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠₁)=𝐻₁(𝑠₁) (𝛽^𝑜𝑝 is order-preserving).
For the last claim, notice that given 𝑠∈𝐿^𝐺1₁ such that 𝐾₂∘𝐻₂∘𝛼(𝑠)=𝛼(𝑠), it follows that 𝛼∘𝐾₁∘𝐻₁(𝑠)=𝛼∘𝐾₁∘𝛽^𝑜𝑝∘𝐻₂∘𝛼(𝑠)=𝐾₂∘𝐻₂∘𝛼(𝑠)=𝛼(𝑠) and thus, 𝐾₁∘𝐻₁(𝑠)=𝑠 (injectivity of 𝛼). Additionally, given 𝑡∈𝐿^𝑀2₂ such that 𝐻₁∘𝐾₁∘𝛽^𝑜𝑝(𝑡)=𝛽^𝑜𝑝(𝑡), it follows that 𝛽^𝑜𝑝∘𝐻₂∘𝐾₂(𝑡)=𝛽^𝑜𝑝∘𝐻₂∘𝛼∘𝐾₁∘𝛽^𝑜𝑝(𝑡)=𝐻₁∘𝐾₁∘𝛽^𝑜𝑝(𝑡)=𝛽^𝑜𝑝(𝑡) and thus, 𝐻₂∘𝐾₂(𝑡)=𝑡 (injectivity of 𝛽^𝑜𝑝).

Theorem 46. Let 𝒦−→₁^𝑓𝒦₂ be a morphism of the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 and let 𝑠₁∈𝐿^𝐺1₁, 𝑡₁∈𝐿^𝑀1₁. (1)If both 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ and 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ are injective, then (𝑠₁,𝑡₁) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal concept of 𝒦₂. (2)If one of 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂, 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ is injective, then (𝑠₁,𝑡₁) is a formal protoconcept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal protoconcept of 𝒦₂. (3)If one of 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂, 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ is order-preserving and, moreover, one of the maps is order-reflecting, then (𝑠₁,𝑡₁) is a formal preconcept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal preconcept of 𝒦₂.

Proof. Ad (1). “⇒”: Since 𝐻₁(𝑠₁)=𝑡₁ and 𝐾₁(𝑡₁)=𝑠₁, 𝐻₂∘𝛼(𝑠₁)=𝛽∘𝐻₁(𝑠₁)=𝛽(𝑡₁) and 𝐾₂∘𝛽(𝑡₁)=𝛼∘𝐾₁(𝑡₁)=𝛼(𝑠₁) (no injectivity condition is employed). “⇐”: Since 𝐻₂∘𝛼(𝑠₁)=𝛽(𝑡₁) and 𝐾₂∘𝛽(𝑡₁)=𝛼(𝑠₁), 𝛽∘𝐻₁(𝑠₁)=𝐻₂∘𝛼(𝑠₁)=𝛽(𝑡₁) and 𝛼∘𝐾₁(𝑡₁)=𝐾₂∘𝛽(𝑡₁)=𝛼(𝑠₁) imply 𝐻₁(𝑠₁)=𝑡₁ and 𝐾₁(𝑡₁)=𝑠₁, respectively (both 𝛼 and 𝛽 are injective).
Ad (2). “⇒”: Since 𝐾₁∘𝐻₁(𝑠₁)=𝐾₁(𝑡₁), 𝐾₂∘𝐻₂∘𝛼(𝑠₁)=𝐾₂∘𝛽∘𝐻₁(𝑠₁)=𝛼∘𝐾₁∘𝐻₁(𝑠₁)=𝛼∘𝐾₁(𝑡₁)=𝐾₂∘𝛽(𝑡₁) (no injectivity condition is employed). “⇐’’: If 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ is injective, then 𝐾₂∘𝐻₂∘𝛼(𝑠₁)=𝐾₂∘𝛽(𝑡₁) implies 𝛼∘𝐾₁∘𝐻₁(𝑠₁)=𝐾₂∘𝛽∘𝐻₁(𝑠₁)=𝐾₂∘𝐻₂∘𝛼(𝑠₁)=𝐾₂∘𝛽(𝑡₁)=𝛼∘𝐾₁(𝑡₁) and thus, 𝐾₁∘𝐻₁(𝑠₁)=𝐾₁(𝑡₁). If 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ is injective, then 𝐻₂∘𝐾₂∘𝛽(𝑡₁)=𝐻₂∘𝛼(𝑠₁) implies 𝛽∘𝐻₁∘𝐾₁(𝑡₁)=𝐻₂∘𝛼∘𝐾₁(𝑡₁)=𝐻₂∘𝐾₂∘𝛽(𝑡₁)=𝐻₂∘𝛼(𝑠₁)=𝛽∘𝐻₁(𝑠₁) and thus, 𝐻₁∘𝐾₁(𝑡₁)=𝐻₁(𝑠₁).
Ad (3). “⇒’’: If 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ is order-preserving, then 𝑠₁⩽𝐾₁(𝑡₁) implies 𝛼(𝑠₁)⩽𝛼∘𝐾₁(𝑡₁)=𝐾₂∘𝛽(𝑡₁). If 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ is order-preserving, then 𝑡₁⩽𝐻₁(𝑠₁) yields 𝛽(𝑡₁)⩽𝛽∘𝐻₁(𝑠₁)=𝐻₂∘𝛼(𝑠₁). “⇐”: If 𝐿→−^𝐺1₁^𝛼𝐿^𝐺2₂ is order-reflecting, then 𝛼(𝑠₁)⩽𝐾₂∘𝛽(𝑡₁)=𝛼∘𝐾₁(𝑡₁) implies 𝑠₁⩽𝐾₁(𝑡₁). If 𝐿→−^𝑀1₁^𝛽𝐿^𝑀2₂ is order-reflecting, then 𝛽(𝑡₁)⩽𝐻₂∘𝛼(𝑠₁)=𝛽∘𝐻₁(𝑠₁) yields 𝑡₁⩽𝐻₁(𝑠₁).

The two previous theorems provide crucial tools for defining certain functors between the categories of lattice-valued formal contexts, which is the main topic of the next section.

5. Functorial Relationships between the Categories of Lattice-Valued Formal Contexts

The previous section introduced several categories (partly motivated by the respective ones of [29]), the objects of which are lattice-valued formal contexts. It is the purpose of this section to consider functorial relationships between the new categories, the functors in question being again partly inspired by the respective approach of [29]. As a result, we not only extend (the change from commutative quantales to noncommutative ones and the shift from fixed-basis to variable-basis) and streamline the machinery of Denniston et al., but also provide several new functors (induced by new categories), thereby clarifying the relationships between different frameworks for doing lattice-valued FCA and bringing to light their respective (dis)advantages.

5.1. 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐶_𝑚 versus 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we consider possible functorial links between the categories 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐶_𝑚, on one side, and the categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, on the other.

5.1.1. From 𝐒-𝐅𝐂^𝐶_𝑚 to 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we construct a functorial embedding of a particular subcategory of the category 𝐒-𝐅𝐂^𝐶_𝑚 into the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅. We begin with singling out the subcategory in question.

Definition 47. 𝐒-𝐅𝐂^𝐶_𝑚∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐶_𝑚, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective, whereas the S-morphism 𝐿→−₁^𝜑𝐿₂ preserves ⋀, →_𝑙 and →_𝑟.

Notice that the notation (−)_∗ will usually be used for the constructed subcategories in this paper, the scope of each (−)_∗ being limited to its respective subsection. More important constructions, however, will be distinguished, respectively.

With the category of Definition 47 in hand, we can construct the following functor (the reader is advised to recall our powerset operator notations from Section 2.2).

Theorem 48. There exists the functor 𝐒-𝐅𝐂−−−−−−→^𝐶𝐻_𝑚∗^{𝐶𝑚𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is defined by the formula 𝐻_{𝐶𝑚𝐷𝑎}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)→,(𝛽,𝜑)←𝑜)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (7)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₂∈𝑀₂, 𝐻₂∘𝛼,𝜑^→(𝑠)𝑚₂=_{𝑔2∈𝐺2}𝛼,𝜑^→(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2}𝜑_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚_2(†)=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚_2()††=_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2()†††=𝜑_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚₂=𝜑∘𝐻₁(𝑠)𝛽^𝑜𝑝𝑚₂=𝛽,𝜑^←𝑜∘𝐻₁(𝑠)𝑚₂, (8) where (†) relies on Theorem 6(2), (††) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐶_𝑚, whereas (†††) employs the definition of the category 𝐒-𝐅𝐂^𝐶_𝑚∗.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀1₁ and 𝑔₂∈𝐺₂, 𝛼,𝜑^→∘𝐾₁(𝑡)𝑔₂=𝜑_{𝛼𝑔1=𝑔2}𝐾₁(𝑡)𝑔₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝐼₁𝑔₁,𝑚_1(†)=𝜑_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2()††=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝜑∘𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2()†††=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₂𝛼𝑔₁,𝑚₂=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2}𝜑∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂𝜑∘𝑡∘𝛽^𝑜𝑝𝑔₂=𝐾₂∘𝛽,𝜑^←𝑜(𝑡)𝑔₂, (9) where (†), (††) use the definition of the category 𝐒-𝐅𝐂^𝐶_𝑚∗, whereas (†††) employs the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of the category 𝐒-𝐅𝐂^𝐶_𝑚.

In the notation “𝐻_{𝐶𝑚𝐷𝑎}”, “𝐶𝑚” shows its domain, and “𝐷𝑎” (short for “𝐷𝑀𝑅𝑎”) stands for its codomain. Similar notational conventions will be used in the remainder of this paper but will not be mentioned explicitly again.

In the following, we make the new functor into an embedding. The result depends on a particular subcategory of its domain.

Definition 49. Suppose that the variety 𝐋, of which 𝐒 is a subcategory, extends the variety UQuant of unital quantales. 𝐒-𝐅𝐂^𝐶_𝑚∗∗ (resp., 𝐒-𝐅𝐂^𝐶_𝑚∗•) is the full subcategory of the category 𝐒-𝐅𝐂^𝐶_𝑚∗, whose objects 𝒦=(𝐺,𝑀,𝐿,𝐼) are such that 𝐺 is nonempty (resp., 𝑀 is nonempty) and, moreover, 1_𝐿≠⟂_𝐿.

The notation (−)_∗∗ (resp., (−)_∗•) will be used for many of the subcategories constructed in this paper, the scope of each being limited to its respective subsection. Additionally, more important constructions will be distinguished in a different way. It is also important to emphasize that the algebra requirement of Definition 49 is satisfied by the crisp case (see Definition 31(1)).

Theorem 50. The restriction 𝐻^∗_{𝐶𝑚𝐷𝑎} (resp., 𝐻^•_{𝐶𝑚𝐷𝑎}) of the functor 𝐒-𝐅𝐂−−−−−−→^𝐶𝐻_𝑚∗^{𝐶𝑚𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 to 𝐒-𝐅𝐂^𝐶_𝑚∗∗ (resp., 𝐒-𝐅𝐂^𝐶_𝑚∗•) gives an (in general, nonfull) embedding of the latter category into 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅.

Proof. We consider the case of the functor 𝐻^∗_{𝐶𝑚𝐷𝑎} first and show its faithfulness. Given 𝒦⇉₁^𝑓1_𝑓2𝒦₂ in 𝐒-𝐅𝐂^𝐶_𝑚∗∗ with 𝐻^∗_{𝐶𝑚𝐷𝑎}(𝑓₁)=𝐻^∗_{𝐶𝑚𝐷𝑎}(𝑓₂), it follows that (𝛼₁,𝜑₁)^→=(𝛼₂,𝜑₂)^→, (𝛽₁,𝜑₁)^←𝑜=(𝛽₂,𝜑₂)^←𝑜.
To show that 𝛼₁=𝛼₂ (which we will refer to as 𝛼), notice that given 𝑔₁∈𝐺₁, 𝛼₁,𝜑₁^→𝜒^1𝐿1_𝑔1𝑔₂=𝜑_{1𝛼1𝑔′1=𝑔2}𝜒^1𝐿1_𝑔1𝑔^′₁=𝜑₁1_𝐿1,𝑔₂=𝛼₁𝑔₁⟂_𝐿1=1,otherwise_𝐿2,𝑔₂=𝛼₁𝑔₁⟂_𝐿2,otherwise=𝜒^1𝐿2_𝛼1𝑔1𝑔₂, (10) for every 𝑔₂∈𝐺₂. As a consequence, we obtain that 𝜒^1𝐿2_{{𝛼1(𝑔1)}}=(𝛼₁,𝜑₁)^→(𝜒^1𝐿1_{𝑔1})=(𝛼₂,𝜑₂)^→(𝜒^1𝐿1_{𝑔1})=𝜒^1𝐿2_{{𝛼2(𝑔1)}} and, therefore, 𝛼₁(𝑔₁)=𝛼₂(𝑔₁), by our assumption that 1_𝐿2≠⟂_𝐿2.
To show that 𝛽₁^𝑜𝑝=𝛽₂^𝑜𝑝=𝛽^𝑜𝑝, notice that given 𝑚₂∈𝑀₂, ((𝛽₁,𝜑₁)^←𝑜(𝜒^1𝐿1_{{𝛽1𝑜𝑝(𝑚2)}}))(𝑚₂)=𝜑₁∘𝜒^1𝐿1_{{𝛽1𝑜𝑝(𝑚2)}}∘𝛽₁^𝑜𝑝(𝑚₂)=1_𝐿2. Additionally, 𝛽₂,𝜑₂^←𝑜𝜒^1𝐿1_{𝛽1𝑜𝑝𝑚2}𝑚₂=𝜑₂∘𝜒^1𝐿1_{𝛽1𝑜𝑝𝑚2}∘𝛽₂^𝑜𝑝𝑚₂=1_𝐿2,𝛽₂^𝑜𝑝𝑚₂=𝛽₁^𝑜𝑝𝑚₂⟂_𝐿2,otherwise. (11) Since ((𝛽₁,𝜑₁)^←𝑜(𝜒^1𝐿1_{{𝛽1𝑜𝑝(𝑚2)}}))(𝑚₂)=((𝛽₂,𝜑₂)^←𝑜(𝜒^1𝐿1_{{𝛽1𝑜𝑝(𝑚2)}}))(𝑚₂), it immediately follows that 𝛽₁^𝑜𝑝(𝑚₂)=𝛽₂^𝑜𝑝(𝑚₂), by our assumption that 1_𝐿2≠⟂_𝐿2.
To show that 𝜑₁=𝜑₂=𝜑, notice that given 𝑎₁∈𝐿₁, we can choose some 𝑔₁∈𝐺₁ (𝐺₁ is nonempty). Then ((𝛼,𝜑₁)^→(𝑎₁))(𝛼(𝑔₁))=𝜑₁(⋁_{𝛼(𝑔′1)=𝛼(𝑔1)}𝑎₁(𝑔^′₁))=𝜑₁(𝑎₁) implies 𝜑₁(𝑎₁)=((𝛼,𝜑₁)^→(𝑎₁))(𝛼(𝑔₁))=((𝛼,𝜑₂)^→(𝑎₁))(𝛼(𝑔₁))=𝜑₂(𝑎₁).
Turning the attention to the functor 𝐻^•_{𝐶𝑚𝐷𝑎}, one follows the same steps to obtain 𝛼₁=𝛼₂=𝛼 and 𝛽₁^𝑜𝑝=𝛽₂^𝑜𝑝=𝛽^𝑜𝑝. The only difference is in the proof of 𝜑₁=𝜑₂=𝜑. Given 𝑎₁∈𝐿₁, we choose some 𝑚₂∈𝑀₂ (𝑀₂ is nonempty). It follows then that ((𝛽,𝜑₁)^←𝑜(𝑎₁))(𝑚₂)=𝜑₁∘𝑎₁∘𝛽^𝑜𝑝(𝑚₂)=𝜑₁(𝑎₁) and, therefore, 𝜑₁(𝑎₁)=((𝛽,𝜑₁)^←𝑜(𝑎₁))(𝑚₂)=((𝛽,𝜑₂)^←𝑜(𝑎₁))(𝑚₂)=𝜑₂(𝑎₁).
For the nonfullness claim, we restrict the setting to the crisp case (𝐋=𝐂𝐁𝐀𝐥𝐠 and 𝐒=𝐒₂) and consider the context 𝒦=(𝐺,𝑀,𝐼), where 𝐺=𝑀=|2| and 𝐼=𝐺×𝑀. By Lemma 43, it follows that 𝒫(𝐺)−−−−−^𝐻=𝑀→𝒫(𝑀) and 𝒫(𝑀)−−−−→^𝐾=𝐺𝒫(𝐺). We define a formal context morphism 𝒦−→^𝑓𝒦 of the category 2-𝐅𝐂_𝑎^𝐷𝑀𝑅 by 𝒫(𝐺)−−−−→^𝛼=𝐺𝒫(𝐺) and 𝒫(𝑀)−−−−^𝛽=𝑀→𝒫(𝑀). Since the setting fits both the functor 𝐻^∗_{𝐶𝑚𝐷𝑎} and 𝐻^•_{𝐶𝑚𝐷𝑎}, we consider the case of the former one. Suppose there exists some 𝒦−−→^𝑓′𝒦 in 2-𝐅𝐂^𝐶_𝑚∗∗ such that 𝐻^∗_{𝐶𝑚𝐷𝑎}(𝑓^′)=𝑓. If follows that 𝛼=(𝛼^′)^→ (crisp forward powerset operator) and, therefore, {𝛼^′(⟂)}=(𝛼^′)^→({⟂})=𝛼({⟂})=𝐺, which is an obvious contradiction.

5.1.2. From 𝐒-𝐅𝐂^𝐶_𝑚 to 𝐒-𝐅𝐂^𝐷𝑀𝑅

In this subsection, we construct a functorial embedding of a particular subcategory of the category 𝐒-𝐅𝐂^𝐶_𝑚 into the category 𝐒-𝐅𝐂^𝐷𝑀𝑅. We begin again with singling out the subcategory in question.

Definition 51. 𝐒-𝐅𝐂^𝐶_𝑚∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐶_𝑚, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective, whereas the 𝐒-morphism 𝐿→−₁^𝜑𝐿₂ is an isomorphism.

With the new definition in hand, we can construct the following functor.

Theorem 52. There exists the functor 𝐒-𝐅𝐂−−−−−→^𝐶𝐻_𝑚∗^𝐶𝑚𝐷𝐒-𝐅𝐂^𝐷𝑀𝑅, which is given by 𝐻_𝐶𝑚𝐷(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)→,((𝛽,𝜑)⊢⇢𝑜)𝑜𝑝)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (12)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₁∈𝑀₁, 𝛽,𝜑^⊢⇢𝑜∘𝐻₂∘𝛼,𝜑^→(𝑠)𝑚₁=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1}𝐻₂∘𝛼,𝜑^→(𝑠)𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2}𝛼,𝜑^→(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2}𝜑_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚_2(†)=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚₂=𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝑚_1()††=𝜑^⊢_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝑚_1()††=𝜑^⊢∘𝜑_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚_1()††=_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚₁=𝐻₁(𝑠)𝑚₁, (13) where (†) relies on the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of the category 𝐒-𝐅𝐂^𝐶_𝑚, whereas (††) employs the definition of the category 𝐒-𝐅𝐂^𝐶_𝑚∗ (notice that the inverse of 𝜑 is easily seen to be 𝜑^⊢).
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀2₂ and 𝑔₂∈𝐺₂, 𝛼,𝜑^→∘𝐾₁∘𝛽,𝜑^⊢⇢𝑜(𝑡)𝑔₂=𝜑_{𝛼𝑔1=𝑔2}𝐾₁∘𝛽,𝜑^⊢⇢𝑜(𝑡)𝑔₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝛽,𝜑^⊢⇢𝑜(𝑡)𝑚₁→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^⊢_{𝛽𝑜𝑝𝑚2=𝑚1}𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1𝛽𝑜𝑝𝑚2=𝑚1}𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1𝛽𝑜𝑝𝑚2=𝑚1}𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2(†)=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝜑^⊢∘𝑡𝑚₂→_𝑟𝜑∘𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2()††=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₂𝛼𝑔₁,𝑚₂=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑∘𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2}𝜑∘𝜑^⊢∘𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂(𝑡)𝑔₂, (14) where (†) uses the definition of the category 𝐒-𝐅𝐂^𝐶_𝑚∗, whereas (††) relies on the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of the category 𝐒-𝐅𝐂^𝐶_𝑚.

The category 𝐒-𝐅𝐂^𝐶_𝑚∗∗ (resp., 𝐒-𝐅𝐂^𝐶_𝑚∗•) of Definition 49 provides the following result.

Theorem 53. The restriction 𝐻^∗_𝐶𝑚𝐷 (resp., 𝐻^•_𝐶𝑚𝐷) of the functor 𝐒-𝐅𝐂−−−−−→^𝐶𝐻_𝑚∗^𝐶𝑚𝐷𝐒-𝐅𝐂^𝐷𝑀𝑅 to 𝐒-𝐅𝐂^𝐶_𝑚∗∗ (resp., 𝐒-𝐅𝐂^𝐶_𝑚∗•) gives an (in general, nonfull) embedding of the latter category into 𝐒-𝐅𝐂^𝐷𝑀𝑅.

Proof. Follow the steps of the proof of Theorem 50, where the backward powerset operator (𝛽,𝜑)^←𝑜 is substituted by the forward powerset operator (𝛽,𝜑)^⊢⇢𝑜.

5.1.3. From 𝐒-𝐅𝐂^𝐶 to 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we construct a functorial embedding of a particular subcategory of the category 𝐒-𝐅𝐂^𝐶 into the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅. We start by defining the subcategory in question.

Definition 54. 𝐒-𝐅𝐂^𝐶_∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐶, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective, whereas the 𝐒^𝑜𝑝-morphism 𝐿−−−→₂^𝜑𝑜𝑝𝐿₁ is injective, and, moreover, |𝐿₁|−−−^{𝜑⊢𝑜𝑝}→|𝐿₂| preserves ⋁, →_𝑙 and →_𝑟.

For the new category in hand, in the following, we construct its respective functor.

Theorem 55. There exists the functor 𝐒-𝐅𝐂−−−−→^𝐶_∗^{𝐻𝐶𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is given by 𝐻_𝐶𝐷𝑎(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)⊢⇢,((𝛽,𝜑)⊢⇠𝑜)𝑜𝑝)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (15)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₂∈𝑀₂, 𝐻₂∘𝛼,𝜑^⊢⇢(𝑠)𝑚₂=_{𝑔2∈𝐺2}𝛼,𝜑^⊢⇢(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2}𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚_2(†)=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚_2(†)=_{𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2(†)=𝜑^⊢𝑜𝑝_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2()††=𝜑^⊢𝑜𝑝_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚₂=𝜑^⊢𝑜𝑝∘𝐻₁(𝑠)∘𝛽^𝑜𝑝𝑚₂=𝛽,𝜑^⊢⇠𝑜∘𝐻₁(𝑠)𝑚₂, (16) where (†) uses Definition 54 (notice that 𝜑^⊢𝑜𝑝 is a left inverse to 𝜑^𝑜𝑝), whereas (††) relies on the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐶.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀1₁ and 𝑔₂∈𝐺₂, 𝛼,𝜑^⊢⇢∘𝐾₁(𝑡)𝑔₂=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝐾₁(𝑡)𝑔₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝐼₁𝑔₁,𝑚_1(†)=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2()††=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚₂=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝑚_2(†)=𝜑^⊢𝑜𝑝_{𝑚2∈𝑀2}𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2}𝜑^⊢𝑜𝑝∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2}𝜑^⊢𝑜𝑝∘𝑡∘𝛽^𝑜𝑝𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂𝜑^⊢𝑜𝑝∘𝑡∘𝛽^𝑜𝑝𝑔₂=𝐾₂∘𝛽,𝜑^⊢⇠𝑜(𝑡)𝑔₂, (17) where (†) uses Definition 54, whereas (††) relies on the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐶.

In the following, we make the new functor into an embedding.

Definition 56. Suppose that the variety 𝐋, of which 𝐒 is a subcategory, extends UQuant. 𝐒-𝐅𝐂^𝐶_∗∗ (resp., 𝐒-𝐅𝐂^𝐶_∗•) is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐶_∗, comprising the following data.
Objects. 𝐒-𝐅𝐂^𝐶_∗-objects 𝒦, for which 𝐺 is nonempty (resp., 𝑀 is nonempty) and, moreover, 1_𝐿≠⟂_𝐿.
Morphisms. 𝐒-𝐅𝐂^𝐶_∗-morphisms 𝒦−→₁^𝑓𝒦₂, for which |𝐿₁|−−−^{𝜑⊢𝑜𝑝}→|𝐿₂| is 1-preserving.

The reader should notice the essential difference in the setting of Definition 49, namely, an additional condition on the morphisms of the new category.

Theorem 57. The restriction 𝐻^∗_𝐶𝐷𝑎 (resp., 𝐻^•_𝐶𝐷𝑎) of the functor 𝐒-𝐅𝐂−−−−→^𝐶_∗^{𝐻𝐶𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 to the category 𝐒-𝐅𝐂^𝐶_∗∗ (resp., 𝐒-𝐅𝐂^𝐶_∗•) gives an (in general nonfull) embedding of the latter category into 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅.

Proof. Follow the proof of Theorem 50, with the respective changes in the powerset operators.

5.1.4. From 𝐒-𝐹𝐂^𝐶 to 𝐒-𝐅𝐂^𝐷𝑀𝑅

In this subsection, we construct a functorial embedding of a particular subcategory of the category 𝐒-𝐅𝐂^𝐶 into the category 𝐒-𝐅𝐂^𝐷𝑀𝑅. As before, we begin with singling out the subcategory in question.

Definition 58. 𝐒-𝐅𝐂^𝐶_∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐶, with the same objects, and whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective, whereas the 𝐒^𝑜𝑝-morphism 𝐿−−−→₂^𝜑𝑜𝑝𝐿₁ is an isomorphism.

With the new category in hand, we are ready to define a new functor as follows.

Theorem 59. There exists the functor 𝐒-𝐅𝐂−−−−→^𝐶_∗^𝐻𝐶𝐷𝐒-𝐅𝐂^𝐷𝑀𝑅, which is given by 𝐻_𝐶𝐷(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)⊢⇢,((𝛽,𝜑)→𝑜)𝑜𝑝)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (18)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₁∈𝑀₁, 𝛽,𝜑^→𝑜∘𝐻₂∘𝛼,𝜑^⊢⇢(𝑠)𝑚₁=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1}𝐻₂∘𝛼,𝜑^⊢⇢(𝑠)𝑚₂=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2}𝛼,𝜑^⊢⇢(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2}𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚_2(†)=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚_2(†)=_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑^𝑜𝑝∘𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2()††=_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝜑^𝑜𝑝∘𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2(†)=_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚₂=_{𝛽𝑜𝑝𝑚2=𝑚1𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚_1(†)=_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚₁=𝐻₁(𝑠)𝑚₁, (19) where (†) uses Definition 58, whereas (††) relies on the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐶.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀2₂ and 𝑔₂∈𝐺₂, 𝛼,𝜑^⊢⇢∘𝐾₁∘𝛽,𝜑^→𝑜(𝑡)𝑔₂=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝐾₁∘𝛽,𝜑^→𝑜(𝑡)𝑔₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝛽,𝜑^→𝑜(𝑡)𝑚₁→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^𝑜𝑝_{𝛽𝑜𝑝𝑚2=𝑚1}𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1𝛽𝑜𝑝𝑚2=𝑚1}𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1𝛽𝑜𝑝𝑚2=𝑚1}𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝐼₁𝑔₁,𝛽^𝑜𝑝𝑚_2(†)=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2()††=𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝛼𝑔₁,𝑚_2()††=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝛼𝑔₁,𝑚₂=_{𝛼𝑔1=𝑔2𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚_2()††=_{𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂(𝑡)𝑔₂, (20) where (†) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐶, whereas (††) employs Definition 58.

Similar to Definition 49, one obtains the categories 𝐒-𝐅𝐂^𝐶_∗∗ and 𝐒-𝐅𝐂^𝐶_∗•. Moreover, similar to Theorem 50, one gets the following result.

Theorem 60. The restriction 𝐻^∗_𝐶𝐷 (resp., 𝐻^•_𝐶𝐷) of the functor 𝐒-𝐅𝐂−−−−→^𝐶_∗^𝐻𝐶𝐷𝐒-𝐅𝐂^𝐷𝑀𝑅 to 𝐒-𝐅𝐂^𝐶_∗∗ (resp., 𝐒-𝐅𝐂^𝐶_∗•) gives an (in general nonfull) embedding of the latter category into 𝐒-𝐅𝐂^𝐷𝑀𝑅.

5.1.5. From 𝐒-𝐅𝐂^𝐷𝑀𝑅 to 𝐒-𝐅𝐂^𝐶

In this subsection, we construct functorial embeddings of several subcategories of the category 𝐒-𝐅𝐂^𝐷𝑀𝑅 into the category 𝐒-𝐅𝐂^𝐶 (most of them are motivated by [29]). We begin with defining the first subcategory in question. The reader should note that all the subcategories constructed in this subsection are fixed-basis.

Definition 61. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂_𝔦^𝐷𝑀𝑅 is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the maps 𝐿→−^𝐺1𝛼𝐿^𝐺2, 𝐿−−−→^{𝑀2𝛽𝑜𝑝}𝐿^𝑀1 are injective.

The reader should notice that “𝔦” in 𝐿-𝐅𝐂_𝔦^𝐷𝑀𝑅 stands for “injective.”

Theorem 62. There exists the functor 𝐿-𝐅𝐂−−−−→_𝔦^{𝐻𝐷𝑀𝑅𝔦𝐷𝐶}𝐒-𝐅𝐂^𝐶, which is given by 𝐻^𝔦_𝐷𝐶(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (21)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₂∈𝐿^𝑀2, it follows that 𝐼₁̂(𝑠₁,𝛽^𝑜𝑝(𝑡₂))=⊤_𝐿 iff (𝑠₁,𝛽^𝑜𝑝(𝑡₂)) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal concept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝑡₂)=⊤_𝐿, where the second “iff” uses Theorem 45(3).

In the following, we make the functor of Theorem 62 into an embedding. To achieve the goal, we impose a restriction on the chosen 𝐋-algebra 𝐿.

Definition 63. An 𝐋-algebra 𝐿 is called quasistrictly right-sided (qsrs-algebra, for short) provided that 𝑎⩽(⊤_𝐿→_𝑙𝑎)⊗⊤_𝐿 for every 𝑎∈𝐿.

Since every quantale 𝑄 has the property (⊤_𝑄→_𝑙𝑞)⊗⊤_𝑄⩽𝑞 for every 𝑞∈𝑄 (use Example 21(2)), every qsrs-algebra 𝐿 satisfies (⊤_𝐿→_𝑙𝑎)⊗⊤_𝐿=𝑎 for every 𝑎∈𝐿. Additionally, if 𝑄 is a strictly right-sided quantale (recall Definition 9), then, for every 𝑞∈𝑄, ⊤_𝐿→_𝑙𝑞=𝑞 yields (⊤_𝐿→_𝑙𝑞)⊗⊤_𝐿=⊤_𝐿→_𝑙𝑞=𝑞. As a result, every 𝐋-algebra 𝐿 extending a strictly right-sided quantale 𝑄 is a qsrs-algebra, but not vice versa.

Theorem 64. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−→_𝔦^{𝐻𝐷𝑀𝑅𝔦𝐷𝐶}𝐒-𝐅𝐂^𝐶 is a nonfull embedding.

Proof. It will be enough to show that the functor 𝐻^𝔦_𝐷𝐶 is injective on objects. Given 𝒦₁, 𝒦₂ in 𝐿-𝐅𝐂_𝔦^𝐷𝑀𝑅 such that 𝐻^𝔦_𝐷𝐶(𝒦₁)=𝐻^𝔦_𝐷𝐶(𝒦₂), it follows that (𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)=(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂) and, therefore, 𝐺₁=𝐺₂=𝐺, 𝑀₁=𝑀₂=𝑀 and 𝐼₁̂=𝐼₂̂. To show that 𝐼₁=𝐼₂, we assume that 𝐼₁≠𝐼₂. Then there exist some 𝑔∈𝐺 and 𝑚∈𝑀 such that 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚). As a consequence, we obtain that ⟂_𝐿≠⊤_𝐿. The properties of Galois connections of Theorem 19 imply that 𝐻₁∘𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔})=𝐻₁(𝜒^⊤𝐿_{𝑔}), and, therefore, (𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}),𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal concept of 𝒦₁. As a consequence, 𝐼₁̂(𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}),𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿 and, thus, 𝐼₂̂(𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}),𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿; that is, (𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}),𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal concept of 𝒦₂. It follows that 𝜒^⊤𝐿_{𝑔}⩽𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔})=𝐾₂∘𝐻₁(𝜒^⊤𝐿_{𝑔}) and, therefore, ⊤_𝐿=𝜒^⊤𝐿_{𝑔}(𝑔)⩽(𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}))(𝑔)=(𝐾₂∘𝐻₁(𝜒^⊤𝐿_{𝑔}))(𝑔)=⋀_{𝑚′∈𝑀}((𝐻₁(𝜒^⊤𝐿_{𝑔}))(𝑚^′)→_𝑟𝐼₂(𝑔,𝑚^′⋀))=_{𝑚′∈𝑀}((⋀_{𝑔′∈𝐺}(𝜒^⊤𝐿_{𝑔}(𝑔^′)→_𝑙𝐼₁(𝑔^′,𝑚^′)))→_𝑟𝐼₂(𝑔,𝑚^′))=⋀_{𝑚′∈𝑀}((⊤_𝐿→_𝑙𝐼₁(𝑔,𝑚^′))→_𝑟𝐼₂(𝑔,𝑚^′))⩽(⊤_𝐿→_𝑙𝐼₁(𝑔,𝑚))→_𝑟𝐼₂(𝑔,𝑚). As a result, 𝐼₁(𝑔,𝑚)⩽^(†)(⊤_𝐿→_𝑙𝐼₁(𝑔,𝑚))⊗⊤_𝐿⩽𝐼₂(𝑔,𝑚), where (†) uses the definition of qsrs-algebras, and, therefore, 𝐼₁(𝑔,𝑚)⩽𝐼₂(𝑔,𝑚), which contradicts our assumption 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚).
To show that the functor is nonfull, we let 𝐋=𝐐𝐮𝐚𝐧𝐭 and 𝐿=([0,1],⋁,⋀), that is, taking the unit interval [0,1], with its standard lattice-theoretic structure. Since 𝐿 is clearly strictly two-sided, it is quasistrictly right-sided. Define a map |𝐿|→−^𝜑|𝐿| by 𝜑(𝑎)=321⋅𝑎,𝑎∈0,2,121⋅𝑎+21,𝑎∈2,1. (22) It follows that 𝐿→−^𝜑𝐿 is an STSQuant-isomorphism, which is different from 1_𝐿. Define now a lattice-valued formal context 𝒦 as 𝐺=𝑀={∗} and 𝐼=⊤_𝐿. It is easy to see that 𝐻^𝔦_𝐷𝐶(𝒦)−−−−−−−−−−−−−→^{𝑓=(1𝐿𝐺,1𝐿𝑀,𝜑𝑜𝑝)}𝐻^𝔦_𝐷𝐶(𝒦) is an 𝐒-𝐅𝐂^𝐶-morphism, since 𝜑(𝑎)=⊤_𝐿 (resp., 𝜑(𝑎)=⟂_𝐿) iff 𝑎=⊤_𝐿 (resp., 𝑎=⟂_𝐿). Additionally, the morphism is clearly not in the image of the functor 𝐻^𝔦_𝐷𝐶.

To continue, we provide the second of the claimed embeddings. As usual, we begin by singling out a particular fixed-basis subcategory of the category 𝐒-𝐅𝐂^𝐷𝑀𝑅.

Definition 65. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑜𝑟𝑝 is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ satisfy the property that the maps 𝐿→−^𝐺1𝛼𝐿^𝐺2, 𝐿−−−→^{𝑀2𝛽𝑜𝑝}𝐿^𝑀1 are order-preserving.

The reader should notice that “𝑜𝑟𝑝” in 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑜𝑟𝑝 stands for “order-preserving.”

Theorem 66. There exists the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅}_𝑜𝑟𝑝^{𝑜𝑟𝑝𝐷𝐶}𝐒-𝐅𝐂^𝐶, which is given by 𝐻^𝑜𝑟𝑝_𝐷𝐶(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑒𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (23)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₂∈𝐿^𝑀2, it follows that 𝐼₁̂(𝑠₁,𝛽^𝑜𝑝(𝑡₁))=⊤_𝐿 iff (𝑠₁,𝛽^𝑜𝑝(𝑡₁)) is a formal preconcept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal preconcept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝑡₂)=⊤_𝐿, where the second “iff” uses Theorem 45(5).

The machinery of Theorem 64 does well enough in this modified case as well.

Theorem 67. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅}_𝑜𝑟𝑝^{𝑜𝑟𝑝𝐷𝐶}𝐒-𝐅𝐂^𝐶 is a nonfull embedding.

Proof. The proof starts precisely as that of Theorem 64. To show that 𝐼₁=𝐼₂, we again assume the existence of some 𝑔∈𝐺 and 𝑚∈𝑀 such that 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚). The properties of Galois connections (Theorem 19) provide 𝜒^⊤𝐿_{𝑔}⩽𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}), which implies that (𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal preconcept of 𝒦₁. It follows that 𝐼₁̂(𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿 and, therefore, 𝐼₂̂(𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿; that is, (𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal preconcept of 𝒦₂, and, as a result, 𝜒^⊤𝐿_{𝑔}⩽𝐾₂∘𝐻₁(𝜒^⊤𝐿_{𝑔}). Similar to the proof of Theorem 64, one obtains that 𝐼₁(𝑔,𝑚)⩽𝐼₂(𝑔,𝑚), thereby getting a contradiction with the assumption 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚).
The nonfullness part follows the steps of the proof of Theorem 64.

To continue, we show yet another one of the claimed embeddings, which never requires singling out a particular subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅 (but is still fixed-basis).

Theorem 68. There exists the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅𝐷𝐶}𝐒-𝐅𝐂^𝐶, which is given by 𝐻_𝐷𝐶(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑜𝑡𝑜𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (24)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₂∈L^𝑀2, it follows that 𝐼₁̂(𝑠₁,𝛽^𝑜𝑝(𝑡₁))=⊤_𝐿 iff (𝑠₁,𝛽^𝑜𝑝(𝑡₁)) is a formal protoconcept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal protoconcept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝑡₂)=⊤_𝐿, where the second “iff” uses Theorem 45(4).

The machinery of Theorem 64 does well even in this extended setting.

Theorem 69. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅𝐷𝐶}𝐒-𝐅𝐂^𝐶 is a nonfull embedding.

Proof. The proof starts precisely as that of Theorem 64. To show that 𝐼₁=𝐼₂, we again assume the existence of some 𝑔∈𝐺 and 𝑚∈𝑀 such that 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚). Since 𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔})=𝐾₁∘𝐻₁(𝜒^⊤𝐿_{𝑔}), (𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal protoconcept of 𝒦₁. It follows that 𝐼₁̂(𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿 and, therefore, 𝐼₂̂(𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔}))=⊤_𝐿, that is, (𝜒^⊤𝐿_{𝑔},𝐻₁(𝜒^⊤𝐿_{𝑔})) is a formal protoconcept of 𝒦₂, and, as a result, 𝜒^⊤𝐿_{𝑔}⩽𝐾₂∘𝐻₂(𝜒^⊤𝐿_{𝑔})=𝐾₂∘𝐻₁(𝜒^⊤𝐿_{𝑔}). Similar to the proof of Theorem 64, one obtains that 𝐼₁(𝑔,𝑚)⩽𝐼₂(𝑔,𝑚), thereby getting a contradiction with the assumption 𝐼₁(𝑔,𝑚)𝐼₂(𝑔,𝑚).
The nonfullness part follows the steps of the proof of Theorem 64.

Theorems 71 and 72 show the last of the claimed embeddings; this embedding does require a particular subcategory of 𝐿-𝐅𝐂^𝐷𝑀𝑅.

Definition 70. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑟𝑓𝑝 is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ satisfy the property that the maps 𝐿→−^𝐺1𝛼𝐿^𝐺2, 𝐿−−−→^{𝑀2𝛽𝑜𝑝}𝐿^𝑀1 reflect fixed points.

The reader should notice that “𝑟𝑓𝑝” in 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑟𝑓𝑝 stands for “reflect fixed points.”

Theorem 71. There exists the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅}_𝑟𝑓𝑝^{𝑟𝑓𝑝𝐷𝐶}𝐒-𝐅𝐂^𝐶, which is given by 𝐻^𝑟𝑓𝑝_𝐷𝐶(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (25)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₂∈𝐿^𝑀2, it follows that 𝐼₁̂(𝑠₁,𝛽^𝑜𝑝(𝑡₁))=⊤_𝐿 iff (𝑠₁,𝛽^𝑜𝑝(𝑡₁)) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝑡₂) is a formal concept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝑡₂)=⊤_𝐿, where the second “iff” uses Theorem 45(2).

The embedding property of the just introduced functor is easy to get.

Theorem 72. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−→^{𝐻𝐷𝑀𝑅}_𝑟𝑓𝑝^{𝑟𝑓𝑝𝐷𝐶}𝐒-𝐅𝐂^𝐶 is a nonfull embedding.

Proof. The proof follows the steps of that of Theorem 64.

In view of the last claim of Theorem 45, the category 𝐿-𝐅𝐂_𝔦^𝐷𝑀𝑅 is a (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑟𝑓𝑝. Thus, the functor 𝐻^𝔦_𝐷𝑀 is the restriction of the functor 𝐻^𝑟𝑓𝑝_𝐷𝑀 to the category 𝐿-𝐅𝐂_𝔦^𝐷𝑀𝑅.

As a last remark, we notice that in every unital quantale 𝑄, (1_𝑄→_𝑙𝑞)⊗1_𝑄=1_𝑄→_𝑙𝑞=𝑞 for every 𝑞∈𝑄. In particular, all the results in this subsection can be done in a more simple way, assuming that 𝐋 extends the variety UQuant of unital quantales and then making use of the unit 1_𝐿 instead of the top element ⊤_𝐿 of the given 𝐋-algebra 𝐿. In view of the previously mentioned property of unital quantales, the respective functors are then embeddings, without any additional requirement on the 𝐋-algebra 𝐿.

5.1.6. From 𝐒-𝐅𝐂^𝐷𝑀𝑅 to 𝐒-𝐅𝐂^𝐶_𝑚

The reader can easily show (as a simple exercise) that the functors from the previous subsection can be used in case of the categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂^𝐶_𝑚 as well, providing similar results. In particular, all the nonfullness examples work also in the new setting.

5.2. 𝐒-𝐅𝐂^𝐺𝑊 and 𝐒-𝐅𝐂_𝑚^𝐺𝑊 versus 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we consider possible functorial links between the categories 𝐒-𝐅𝐂^𝐺𝑊 and 𝐒-𝐅𝐂_𝑚^𝐺𝑊, on one side, and the categories 𝐒-𝐅𝐂^𝐷𝑀𝑅 and 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, on the other. The obtained functors can be easily made into embeddings, employing the machinery of Section 5.1 related to the functors from the constructed subcategories of the category 𝐒-𝐅𝐂^𝐶(resp., 𝐒-𝐅𝐂^𝐶_𝑚) to the category 𝐒-𝐅𝐂^𝐷𝑀𝑅 (resp., 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅). In particular, assuming that 𝐋 extends the variety UQuant of unital quantales, whereas 𝐒-𝐅𝐂^𝐺𝑊_†∗is the domain of the newly constructed functor (see the results of following subsections), where † is ether “” or “𝑚,” we can define the following two subcategories.

Definition 73. 𝐒-𝐅𝐂^𝐺𝑊_†∗∗ (resp., 𝐒-𝐅𝐂^𝐺𝑊_†∗•) is the full subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊_†∗, whose objects 𝒦 are such that 𝐺 is nonempty (resp., 𝑀 is nonempty) and, moreover, 1_𝐿≠⟂_𝐿.

Employing the technique of Section 5.1, we can easily get that the restriction of the respective constructed functor to the previously mentioned two categories provides two embeddings. To save the space, therefore, we will not mention the just discussed developments explicitly.

5.2.1. From 𝐒-𝐅𝐂_𝑚^𝐺𝑊 to 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we construct a functor from a particular subcategory of the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊 to the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅. To begin with, we provide the definition of the subcategory in question.

Definition 74. 𝐒-𝐅𝐂^𝐺𝑊_𝑚∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂ and 𝑀→−₁^𝛽𝑀₂ are surjective, whereas the 𝐒-morphism 𝐿→−₁^𝜑𝐿₂ preserves ⋀, →_𝑙 and →_𝑟.

For the new category in hand, we can construct the following functor.

Theorem 75. There exists the functor 𝐒-𝐅𝐂−−−−−−→^𝐻𝐺𝑊_𝑚∗^{𝐺𝑚𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is defined by the formula 𝐻_{𝐺𝑚𝐷𝑎}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−→₁^{((𝛼,𝜑)→,(𝛽,𝜑)→)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (26)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₂∈𝑀₂, 𝛽,𝜑^→∘𝐻₁(𝑠)𝑚₂=𝜑_{𝛽𝑚1=𝑚2}𝐻₁(𝑠)𝑚₁=𝜑_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚_1(†)=_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝑚_1()††=_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝛽𝑚₁=_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚_2(†)=_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2}𝜑_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2}𝛼,𝜑^→(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=𝐻₂∘𝛼,𝜑^→(𝑠)𝑚₂, (27) where (†) uses Definition 74, whereas (††) employs the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂_𝑚^𝐺𝑊.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀1₁ and 𝑔₂∈𝐺₂, 𝐾₂∘𝛽,𝜑^→(𝑡)𝑔₂=_{𝑚2∈𝑀2}𝛽,𝜑^→(𝑡)𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚2∈𝑀2}𝜑_{𝛽𝑚1=𝑚2}𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚2∈𝑀2𝛽𝑚1=𝑚2}𝜑∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚2∈𝑀2𝛽𝑚1=𝑚2}𝜑∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚1∈𝑀1}𝜑∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝛽_1(†)𝑚=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑∘𝑡𝑚₁→_𝑟𝐼₂𝛼𝑔₁,𝛽_1()††𝑚=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑∘𝑡𝑚₁→_𝑟𝜑∘𝐼₁𝑔₁,𝑚_1(†)=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑_{𝛼𝑔1=𝑔2}𝐾₁(𝑡)𝑔₁=𝛼,𝜑^→∘𝐾₁(𝑡)𝑔₂, (28) where (†) uses Definition 74, whereas (††) employs the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂_𝑎^𝐺𝑊.

5.2.2. From 𝐒-𝐅𝐂_𝑚^𝐺𝑊to 𝐒-𝐅𝐂^𝐷𝑀𝑅

In this subsection, we construct a functor from a particular subcategory of the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊 to the category 𝐒-𝐅𝐂^𝐷𝑀𝑅. To begin with, we provide the definition of the subcategory in question.

Definition 76. 𝐒-𝐅𝐂^𝐺𝑊_𝑚∗ is the (nonfull) subcategory of the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂ and 𝑀→−₁^𝛽𝑀₂ are surjective, whereas the 𝐒-morphism 𝐿→−₁^𝜑𝐿₂ is an isomorphism.

Theorem 77. There exists the functor 𝐒-𝐅𝐂−−−−−→^𝐻𝐺𝑊_𝑚∗^𝐺𝑚𝐷𝐒-𝐅𝐂^𝐷𝑀𝑅, which is defined by the formula 𝐻_𝐺𝑚𝐷(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)→,((𝛽,𝜑)⊢⇠)𝑜𝑝)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (29)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₁∈𝑀₁, 𝛽,𝜑^⊢⇠∘𝐻₂∘𝛼,𝜑^→(𝑠)𝑚₁=𝜑^⊢∘𝐻₂𝛼,𝜑^→(𝑠)∘𝛽𝑚₁=𝜑^⊢_{𝑔2∈𝐺2}𝛼,𝜑^→(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^⊢_{𝑔2∈𝐺2}𝜑_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^⊢_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^⊢_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^⊢_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝛽_1(†)𝑚=𝜑^⊢_{𝑔1∈𝐺1}𝜑∘𝑠𝑔₁→_𝑙𝜑∘𝐼₁𝑔₁,𝑚_1()††=𝜑^⊢∘𝜑_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚_1()††=_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚₁=𝐻₁(𝑠)𝑚₁, (30) where (†) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂_𝑚^𝐺𝑊, whereas (††) employs Definition 76.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀2₂ and 𝑔₂∈𝐺₂, 𝛼,𝜑^→∘𝐾₁∘𝛽,𝜑^⊢⇠(𝑡)𝑔₂=𝛼,𝜑^→∘𝐾₁𝜑^⊢∘𝑡∘𝛽𝑔₂=𝜑_{𝛼𝑔1=𝑔2}𝐾₁𝜑^⊢∘𝑡∘𝛽𝑔₁=𝜑_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^⊢∘𝑡∘𝛽𝑚₁→_𝑟𝐼₁𝑔₁,𝑚_1(†)=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑∘𝜑^⊢∘𝑡∘𝛽𝑚₁→_𝑟𝜑∘𝐼₁𝑔₁,𝑚_1()††=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑∘𝜑^⊢∘𝑡∘𝛽𝑚₁→_𝑟𝐼₂𝛼𝑔₁,𝛽_1(†)𝑚=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡∘𝛽𝑚₁→_𝑟𝐼₂𝛼𝑔₁,𝛽𝑚₁=_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡∘𝛽𝑚₁→_𝑟𝐼₂𝑔₂,𝛽_1(†)𝑚=_{𝑚1∈𝑀1}𝑡∘𝛽𝑚₁→_𝑟𝐼₂𝑔₂,𝛽_1(†)𝑚=_{𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂(𝑡)𝑔₂, (31) where (†) uses Definition 76, whereas (††) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂_𝑚^𝐺𝑊.

5.2.3. From 𝐒-𝐅𝐂^𝐺𝑊 to 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅

In this subsection, we construct a functor from a particular subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊 to the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅. As before, we start with singling out the subcategory in question.

Definition 78. 𝐒-𝐅𝐂_∗^𝐺𝑊is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀→−₁^𝛽𝑀₂ are surjective, whereas the 𝐒^𝑜𝑝-morphism 𝐿−−−→₂^𝜑𝑜𝑝𝐿₁ is injective and, moreover, the map |𝐿₁|−−−^{𝜑⊢𝑜𝑝}→|𝐿₂| preserves ⋁, →_𝑙 and →_𝑟.

Theorem 79. There exists the functor 𝐒-𝐅𝐂−−−−→_∗^{𝐻𝐺𝑊𝐺𝐷𝑎}𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is defined by 𝐻_𝐺𝐷𝑎(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)⊢⇢,(𝛽,𝜑)⊢⇢)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (32)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₂∈𝑀₂, 𝛽,𝜑^⊢⇢∘𝐻₁(𝑠)𝑚₂=𝜑^⊢𝑜𝑝_{𝛽𝑚1=𝑚2}𝐻₁(𝑠)𝑚₁=𝜑^⊢𝑜𝑝_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚_1(†)=𝜑^⊢𝑜𝑝_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝛽𝑚₁=𝜑^⊢𝑜𝑝_{𝛽𝑚1=𝑚2𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2()††=𝜑^⊢𝑜𝑝_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2(††)=_{𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝑚_2()††=_{𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚_2(††)=_{𝑔2∈𝐺2}𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝑚₂=_{𝑔2∈𝐺2}𝛼,𝜑^⊢⇢(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝑚₂=𝐻₂∘𝛼,𝜑^⊢⇢(𝑠)𝑚₂, (33) where (†) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐺𝑊, whereas (††) employs Definition 78.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀1₁ and 𝑔₂∈𝐺₂, 𝐾₂∘𝛽,𝜑^⊢⇢(𝑡)𝑔₂=_{𝑚2∈𝑀2}𝛽,𝜑^⊢⇢(𝑡)𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚2∈𝑀2}𝜑^⊢𝑜𝑝_{𝛽𝑚1=𝑚2}𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚_2(†)=_{𝑚2∈𝑀2𝛽𝑚1=𝑚2}𝜑^⊢𝑜𝑝∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚2∈𝑀2𝛽𝑚1=𝑚2}𝜑^⊢𝑜𝑝∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝑚₂=_{𝑚1∈𝑀1}𝜑^⊢𝑜𝑝∘𝑡𝑚₁→_𝑟𝐼₂𝑔₂,𝛽_1(†)𝑚=_{𝑚1∈𝑀1}𝜑^⊢𝑜𝑝∘𝑡𝑚₁→_𝑟𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝛽_1(†)𝑚=𝜑^⊢𝑜𝑝_{𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝛽_1(†)𝑚=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝛽𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝛽_1(††)𝑚=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝑡𝑚₁→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝐾₁(𝑡)𝑔₁=𝛼,𝜑^⊢⇢∘𝐾₁(𝑡)𝑔₂, (34) where (†) uses Definition 78, whereas (††) employs the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐺𝑊.

5.2.4. From 𝐒-𝐅𝐂^𝐺𝑊 to 𝐒-𝐅𝐂^𝐷𝑀𝑅

In this subsection, we construct a functor from a particular subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊 to the category 𝐒-𝐅𝐂^𝐷𝑀𝑅. As before, we begin with singling out the subcategory in question.

Definition 80. 𝐒-𝐅𝐂_∗^𝐺𝑊 is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ are such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀→−₁^𝛽𝑀₂ are surjective, whereas the 𝐒^𝑜𝑝-morphism 𝐿−−−→₂^𝜑𝑜𝑝𝐿₁ is an isomorphism.

Theorem 81. There exists the functor 𝐒-𝐅𝐂−−−−→_∗^{𝐻𝐺𝑊𝐺𝐷}𝐒-𝐅𝐂^𝐷𝑀𝑅, which is defined by 𝐻_𝐺𝐷(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−−−−→₁^{((𝛼,𝜑)⊢⇢,((𝛽,𝜑)←)𝑜𝑝)}𝒦₂.

Proof. To show that the functor is correct on morphisms, we verify commutativity of the diagrams (35)
For the left-hand diagram, notice that given 𝑠∈𝐿^𝐺1₁ and 𝑚₁∈𝑀₁, 𝛽,𝜑^←∘𝐻₂∘𝛼,𝜑^⊢⇢(𝑠)𝑚₁=𝜑^𝑜𝑝∘𝐻₂𝛼,𝜑^⊢⇢(𝑠)∘𝛽𝑚₁=𝜑^𝑜𝑝_{𝑔2∈𝐺2}𝛼,𝜑^⊢⇢(𝑠)𝑔₂→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^𝑜𝑝_{𝑔2∈𝐺2}𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝛽_1(†)𝑚=𝜑^𝑜𝑝_{𝑔2∈𝐺2𝛼𝑔1=𝑔2}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝑔₂,𝛽𝑚₁=𝜑^𝑜𝑝_{𝑔1∈𝐺1}𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝐼₂𝛼𝑔₁,𝛽_1(†)𝑚=_{𝑔1∈𝐺1}𝜑^𝑜𝑝∘𝜑^⊢𝑜𝑝∘𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝛽_1(†)𝑚=_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝛽_1(††)𝑚=_{𝑔1∈𝐺1}𝑠𝑔₁→_𝑙𝐼₁𝑔₁,𝑚₁=𝐻₁(𝑠)𝑚₁, (36) where (†) uses Definition 80, whereas (††) employs the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐺𝑊.
For the right-hand diagram, notice that given 𝑡∈𝐿^𝑀2₂ and 𝑔₂∈𝐺₂, 𝛼,𝜑^⊢⇢∘𝐾₁∘𝛽,𝜑^←(𝑡)𝑔₂=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2}𝐾₁∘𝛽,𝜑^←(𝑡)𝑔₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝛽,𝜑^←(𝑡)𝑚₁→_𝑟𝐼₁𝑔₁,𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^𝑜𝑝∘𝑡∘𝛽𝑚₁→_𝑟𝐼₁𝑔₁,𝑚_1(†)=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^𝑜𝑝∘𝑡∘𝛽𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝛼𝑔₁,𝛽𝑚₁=𝜑^⊢𝑜𝑝_{𝛼𝑔1=𝑔2𝑚1∈𝑀1}𝜑^𝑜𝑝∘𝑡∘𝛽𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝛽_1(††)𝑚=𝜑^⊢𝑜𝑝_{𝑚1∈𝑀1}𝜑^𝑜𝑝∘𝑡∘𝛽𝑚₁→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝛽_1()††𝑚=𝜑^⊢𝑜𝑝_{𝑚2∈𝑀2}𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝑚_2(††)=_{𝑚2∈𝑀2}𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝑡𝑚₂→_𝑟𝜑^⊢𝑜𝑝∘𝜑^𝑜𝑝∘𝐼₂𝑔₂,𝑚_2()††=_{𝑚2∈𝑀2}𝑡𝑚₂→_𝑟𝐼₂𝑔₂,𝑚₂=𝐾₂(𝑡)𝑔₂, (37) where (†) uses the fact that 𝒦−→₁^𝑓𝒦₂ is a morphism of 𝐒-𝐅𝐂^𝐺𝑊, whereas (††) employs Definition 80.

5.2.5. From 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 to 𝐒-𝐅𝐂_𝑚^𝐺𝑊

In this subsection, we construct functorial embeddings of several subcategories of the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅 into the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊. We begin with singling out the first subcategory in question. The reader should notice that similar to Section 5.1.5, all the subcategories constructed in the current one are fixed-basis.

Definition 82. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑎𝔦 is the (nonfull) subcategory of the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the maps 𝐿→−^𝐺1𝛼𝐿^𝐺2, 𝐿→−^𝑀1𝛽𝐿^𝑀2 are injective.

Theorem 83. There exists the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_𝑎𝔦^{𝔦𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊, which is defined by the formula 𝐻^𝔦_{𝐷𝑎𝐺𝑚}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (38)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₁∈𝐿^𝑀1, it follows that 𝐼₁̂(𝑠₁,𝑡₁)=⊤_𝐿 iff (𝑠₁,𝑡₁) is a formal concept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal concept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝛽(𝑡₁))=⊤_𝐿, where the second “iff’’ uses Theorem 46(1).

To make the functor into an embedding, one uses the machinery of Theorem 64.

Theorem 84. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_𝑎𝔦^{𝔦𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊 is a nonfull embedding.

Proof. Follow the steps of the proof of Theorem 64.

To continue, we provide the second of the claimed embeddings. We begin with singling out the subcategory of the category 𝐒-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is suitable for the occasion.

Definition 85. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑎𝔦𝛼 (resp., 𝐿-𝐅𝐂^𝐷𝑀𝑅_𝑎𝔦𝛽) is the (nonfull) subcategory of the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the map 𝐿→−^𝐺1𝛼𝐿^𝐺2 (resp., 𝐿→−^𝑀1𝛽𝐿^𝑀2) is injective.

Theorem 86. Let † be either 𝛼 or 𝛽. There exists the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_𝑎𝔦†^{𝔦†𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊, which is given by 𝐻^𝔦†_{𝐷𝑎𝐺𝑚}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑜𝑡𝑜𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (39)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₁∈𝐿^𝑀1, it follows that 𝐼₁̂(𝑠₁,𝑡₁)=⊤_𝐿 iff (𝑠₁,𝑡₁) is a formal protoconcept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal protoconcept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝛽(𝑡₁))=⊤_𝐿, where the second “iff’’ uses Theorem 46(2).

Theorem 87. Let † be either 𝛼 or 𝛽. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_𝑎𝔦†^{𝔦†𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊 is a nonfull embedding.

Proof. Follow the steps of the proof of Theorem 64.

To continue, we show the last one of the claimed embeddings.

Definition 88. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐷𝑀𝑅_{𝑎𝑜𝑒𝛼} (resp., 𝐿-𝐅𝐂^𝐷𝑀𝑅_{𝑎𝑜𝑒𝛽}) is the (nonfull) subcategory of the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the map 𝐿→−^𝐺1𝛼𝐿^𝐺2 (resp., 𝐿→−^𝑀1𝛽𝐿^𝑀2) is an order embedding (i.e., both order-preserving and order-reflecting).

Theorem 89. Let † be either 𝛼 or 𝛽. There exists the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_{𝑎𝑜𝑒†}^{𝑜𝑒†𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊, which is given by 𝐻^𝑜𝑒†_{𝐷𝑎𝐺𝑚}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐿,𝐼₁̂)−−−−−−→^{(𝛼,𝛽,1𝐿)}(𝐿^𝐺2,𝐿^𝑀2,𝐿,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑒𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (40)

Proof. We have to show that the functor is correct on morphisms. Given 𝑠₁∈𝐿^𝐺1 and 𝑡₁∈𝐿^𝑀1, it follows that 𝐼₁̂(𝑠₁,𝑡₁)=⊤_𝐿 iff (s₁,𝑡₁) is a formal preconcept of 𝒦₁ iff (𝛼(𝑠₁),𝛽(𝑡₁)) is a formal preconcept of 𝒦₂ iff 𝐼₂̂(𝛼(𝑠₁),𝛽(𝑡₁))=⊤_𝐿, where the second “iff’’ uses Theorem 46(3).

Theorem 90. Let † be either 𝛼 or 𝛽. For every qsrs-algebra 𝐿, the functor 𝐿-𝐅𝐂−−−−−−→^{𝐻𝐷𝑀𝑅}_{𝑎𝑜𝑒†}^{𝑜𝑒†𝐷𝑎𝐺𝑚}𝐒-𝐅𝐂_𝑚^𝐺𝑊 is a nonfull embedding.

Proof. Follow the steps of the proof of Theorem 64.

As a last remark, we notice that the case of 𝐋 extending the variety UQuant simplifies the previously mentioned results (see the remark at the end of Section 5.1.5).

5.3. 𝐒-𝐅𝐂^𝐺𝑊 and 𝐒-𝐅𝐂_𝑚^𝐺𝑊 versus 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐶_𝑚

In this subsection, we consider possible functorial links between the categories 𝐒-𝐅𝐂^𝐺𝑊 and 𝐒-𝐅𝐂_𝑚^𝐺𝑊, on one side, and the categories 𝐒-𝐅𝐂^𝐶 and 𝐒-𝐅𝐂^𝐶_𝑚, on the other.

5.3.1. From 𝐒-𝐅𝐂_𝑚^𝐺𝑊 to 𝐒-𝐅𝐂^𝐶_𝑚

In this subsection, we construct a functorial isomorphism between particular subcategories of the categories 𝐒-𝐅𝐂_𝑚^𝐺𝑊 and 𝐒-𝐅𝐂^𝐶_𝑚, respectively.

Definition 91. 𝐒-𝐅𝐂^𝐺𝑊_𝑚∗ (resp., 𝐒-𝐅𝐂^𝐶_𝑚∗) is the (nonfull) subcategory of the category 𝐒-𝐅𝐂_𝑚^𝐺𝑊 (resp., 𝐒-𝐅𝐂^𝐶_𝑚), with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the map 𝑀→−₁^𝛽𝑀₂ (resp., 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁) is bijective.

The categories of Definition 91 give rise to the following functors.

Theorem 92. There exist two functors (1)𝐒-𝐅𝐂−−−−−−→^𝐻𝐺𝑊_𝑚∗^{𝐺𝑚𝐶𝑚}𝐒-𝐅𝐂^𝐶_𝑚∗, which is given by 𝐻_{𝐺𝑚𝐶𝑚}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−→₁^{(𝛼,(𝛽−1)𝑜𝑝,𝜑)}𝒦₂,(2)𝐒-𝐅𝐂−−−−−−→^𝐶𝐻_𝑚∗^{𝐶𝑚𝐺𝑚}𝐒-𝐅𝐂^𝐺𝑊_𝑚∗, which is given by 𝐻_{𝐶𝑚𝐺𝑚}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−→₁^{(𝛼,(𝛽𝑜𝑝)−1,𝜑)}𝒦₂, for which 𝐻_{𝐺𝑚𝐶𝑚}∘𝐻_{𝐶𝑚𝐺𝑚}=1_{𝐒-𝐅𝐂}^𝐶_𝑚∗ and 𝐻_{𝐶𝑚𝐺𝑚}∘𝐻_{𝐺𝑚𝐶𝑚}=1_{𝐒-𝐅𝐂}^𝐺𝑊_𝑚∗.

Proof. To show that the functor 𝐻_{𝐺𝑚𝐶𝑚} is correct on morphisms, notice that given 𝑔₁∈𝐺₁ and 𝑚₂∈𝑀₂, it follows that 𝜑∘𝐼₁(𝑔₁,𝛽⁻¹(𝑚₂))=𝐼₂(𝛼(𝑔₁),𝛽∘𝛽⁻¹(𝑚₂))=𝐼₂(𝛼(𝑔₁),𝑚₂). Additionally, to show that the functor 𝐻_{𝐶𝑚𝐺𝑚} is correct on morphisms, notice that given 𝑔₁∈𝐺₁ and 𝑚₁∈𝑀₁, it follows that 𝜑∘𝐼₁(𝑔₁,𝑚₁)=𝜑∘𝐼₁(𝑔₁,𝛽^𝑜𝑝(𝑚₂))=𝐼₂(𝛼(𝑔₁),𝑚₂)=𝐼₂(𝛼(𝑔₁),(𝛽^𝑜𝑝)⁻¹(𝑚₁)).

5.3.2. From 𝐒-𝐅𝐂^𝐺𝑊 to 𝐒-𝐅𝐂^𝐶

In this subsection, we construct an isomorphism between particular subcategories of the categories 𝐒-𝐅𝐂^𝐺𝑊 and 𝐒-𝐅𝐂^𝐶, respectively.

Definition 93. 𝐒-𝐅𝐂_∗^𝐺𝑊 (resp., 𝐒-𝐅𝐂^𝐶_∗) is the (nonfull) subcategory of the category 𝐒-𝐅𝐂^𝐺𝑊 (resp., 𝐒-𝐅𝐂^𝐶), with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the map 𝑀→−₁^𝛽𝑀₂ (resp., 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁) is bijective.

The categories of Definition 93 give rise to the following functors.

Theorem 94. There exist two functors (1)𝐒-𝐅𝐂−−−−→_∗^{𝐻𝐺𝑊𝐺𝐶}𝐒-𝐅𝐂^𝐶_∗, which is given by 𝐻_𝐺𝐶(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−→₁^{(𝛼,(𝛽−1)𝑜𝑝,𝜑)}𝒦₂,(2)𝐒-𝐅𝐂−−−−→^𝐶_∗^𝐻𝐶𝐺𝐒-𝐅𝐂_∗^𝐺𝑊, which is given by 𝐻_𝐶𝐺(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−→₁^{(𝛼,(𝛽𝑜𝑝)−1,𝜑)}𝒦₂, for which 𝐻_𝐺𝐶∘𝐻_𝐶𝐺=1_{𝐒-𝐅𝐂}^𝐶_∗ and 𝐻_𝐶𝐺∘𝐻_𝐺𝐶=1_{𝐒-𝐅𝐂∗}^𝐺𝑊.

Proof. To show that the functor 𝐻_𝐺𝐶 is correct on morphisms, notice that given 𝑔₁∈𝐺₁ and 𝑚₂∈𝑀₂, it follows that 𝐼₁(𝑔₁,𝛽⁻¹(𝑚₂))=𝜑^𝑜𝑝∘𝐼₂(𝛼(𝑔₁),𝛽∘𝛽⁻¹(𝑚₂))=𝜑^𝑜𝑝∘𝐼₂(𝛼(𝑔₁),𝑚₂). Additionally, to show that the functor 𝐻_𝐶𝐺 is correct on morphisms, notice that given 𝑔₁∈𝐺₁ and 𝑚₁∈𝑀₁, it follows that 𝐼₁(𝑔₁,𝑚₁)=𝐼₁(𝑔₁,𝛽^𝑜𝑝(𝑚₂))=𝜑^𝑜𝑝∘𝐼₂(𝛼(𝑔₁),𝑚₂)=𝐼₂(𝛼(𝑔₁),(𝛽^𝑜𝑝)⁻¹(𝑚₁)).

5.4. 𝐿-𝐅𝐂^𝐶 versus 𝐿-𝐅𝐂^𝐷𝑀𝑅

In this subsection, we make our framework closer to that of Denniston et al. [29], that is, consider a fixed-basis approach to lattice-valued FCA. We construct several functors between the categories 𝐿-𝐅𝐂^𝐶 and 𝐿-𝐅𝐂^𝐷𝑀𝑅 and show that they fail to provide an adjoint situation between the categories in question.

5.4.1. From 𝐿-𝐅𝐂^𝐶 to 𝐿-𝐅𝐂^𝐷𝑀𝑅

This subsection provides a functor from the category 𝐿-𝐅𝐂^𝐶 to the category 𝐿-𝐅𝐂^𝐷𝑀R. More precisely, Theorems 52 and 53 (or, also, Theorems 59 and 60) give rise to the following result.

Definition 95. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐶_∗ is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐶, with the same objects and with morphisms 𝒦−→₁^𝑓𝒦₂ such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective.

Theorem 96. There exists the functor 𝐿-𝐅𝐂−−−−−→^𝐶_∗^{𝐻𝐿𝐶𝐿𝐷}𝐿-𝐅𝐂^𝐷𝑀𝑅, which is given by 𝐻_{𝐿𝐶𝐿𝐷}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−−−−→₁^{(𝛼→𝐿,((𝛽𝑜𝑝)→𝐿)𝑜𝑝)}𝒦₂. If ⊤_𝐿≠⟂_𝐿, then 𝐻_{𝐿𝐶𝐿𝐷} is a nonfull embedding.

Proof. It is enough to show that the functor is faithful (we do not require 𝐿 to have the unit).
Given some 𝐿-𝐅𝐂^𝐶-morphisms 𝒦⇉₁^𝑓1_𝑓2𝒦₂ such that 𝐻_{𝐿𝐶𝐿𝐷}(𝑓₁)=𝐻_{𝐿𝐶𝐿𝐷}(𝑓₂), it follows that (𝛼₁^→_𝐿,((𝛽₁^𝑜𝑝)^→_𝐿)^𝑜𝑝)=(𝛼₂^→_𝐿,((𝛽₂^𝑜𝑝)^→_𝐿)^𝑜𝑝) and, therefore, 𝛼₁^→_𝐿=𝛼₂^→_𝐿, (𝛽₁^𝑜𝑝)^→_𝐿=(𝛽₂^𝑜𝑝)^→_𝐿. To show that 𝛼₁=𝛼₂, notice that given 𝑔₁∈𝐺₁, 𝛼₁^→_𝐿𝜒^⊤𝐿_𝑔1𝑔₂=_{𝛼1𝑔′1=𝑔2}𝜒^⊤𝐿_𝑔1𝑔^′₁=⊤_𝐿,𝑔₂=𝛼₁𝑔₁⟂_𝐿,otherwise=𝜒^⊤𝐿_𝛼1𝑔1𝑔₂ (41) for every 𝑔₂∈𝐺₂. As a consequence, 𝜒^⊤𝐿_{{𝛼1(𝑔1)}}=𝛼₁^→_𝐿(𝜒^⊤𝐿_{𝑔1})=𝛼₂^→_𝐿(𝜒^⊤𝐿_{𝑔1})=𝜒^⊤𝐿_{{𝛼2(𝑔1)}}. The assumption ⊤_𝐿≠⟂_𝐿 then gives 𝛼₁(𝑔₁)=𝛼₂(𝑔₁). Similar technique can be employed to show that 𝛽₁^𝑜𝑝=𝛽₂^𝑜𝑝.

5.4.2. From 𝐿-𝐅𝐂^𝐷𝑀𝑅 to 𝐿-𝐅𝐂^𝐶

This subsection shows functorial embeddings of some subcategories of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅 into the category 𝐿-𝐅𝐂^𝐶. The results of Section 5.1.5 (with their respective notations) suggest the following.

Theorem 97. Every qsrs-algebra 𝐿 provides the following nonfull embeddings.(1)𝐿-𝐅𝐂−−−−−→_𝔦^{𝐻𝐷𝑀𝑅𝔦𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶 given by 𝐻^𝔦_{𝐿𝐷𝐿𝐶}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐼₁̂)−−−−→^(𝛼,𝛽)(𝐿^𝐺2,𝐿^𝑀2,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (42)(2)𝐿-𝐅𝐂−−−−−→^{𝐻𝐷𝑀𝑅}_𝑟𝑓𝑝^{𝑟𝑓𝑝𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶 given by 𝐻^𝑟𝑓𝑝_{𝐿𝐷𝐿𝐶}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐼₁̂)−−−−→^(𝛼,𝛽)(𝐿^𝐺2,𝐿^𝑀2,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (43)(3)𝐿-𝐅𝐂−−−−−→^{𝐻𝐷𝑀𝑅}_𝑜𝑟𝑝^{𝑜𝑟𝑝𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶 given by 𝐻^𝑜𝑟𝑝_{𝐿𝐷𝐿𝐶}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐼₁̂)−−−−→^(𝛼,𝛽)(𝐿^𝐺2,𝐿^𝑀2,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑒𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (44)(4)𝐿-𝐅𝐂−−−−−→^{𝐻𝐷𝑀𝑅𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶 given by 𝐻_{𝐿𝐷𝐿𝐶}(𝒦−→₁^𝑓𝒦₂)=(𝐿^𝐺1,𝐿^𝑀1,𝐼₁̂)−−−−→^(𝛼,𝛽)(𝐿^𝐺2,𝐿^𝑀2,𝐼₂̂), where 𝐼_𝑗̂()=⊤𝑠,𝑡^𝐿,(𝑠,𝑡)𝑖𝑠𝑎𝑓𝑜𝑟𝑚𝑎𝑙𝑝𝑟𝑜𝑡𝑜𝑐𝑜𝑛𝑐𝑒𝑝𝑡𝑜𝑓𝒦_𝑗⟂_𝐿,𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒. (45)

Proof. The only thing that requires verification is nonfullness of the embeddings. We provide the explicit proof for the last item, which can be easily adapted for each of the remaining ones.
Consider the crisp case 𝐋=𝐂𝐁𝐀𝐥𝐠, 𝐿=2, and let 𝐺=𝑀=|𝐿|, 𝐼=𝐺×𝑀. We define (cf. Lemma 43) the 𝐿-𝐅𝐂^𝐷𝑀𝑅-object 𝒦=(𝐺,𝑀,𝐼) where 𝐼=⊤_𝐿, and thus, 𝐻=⊤_𝐿 and 𝐾=⊤_𝐿. For every 𝑆∈𝒫(𝐺) and every 𝑇∈𝒫(𝑀), 𝐻∘𝐾(𝑇)=𝑀=𝐻(𝑆) and thus, (𝑆,𝑇) is a formal protoconcept of 𝒦. Thus, 𝐼̂=⊤_𝐿, and, therefore, every pair of maps 𝒫(𝐺)→−^𝛼𝒫(𝐺), 𝒫(𝑀)−−−→^𝛽𝑜𝑝𝒫(𝑀) is then an 𝐿-𝐅𝐂^𝐶-morphism (𝐻_{𝐿𝐷𝐿𝐶}(𝒦)=(𝒫(𝐺),𝒫(𝑀),𝐼)̂)−−−−→^(𝛼,𝛽)𝐻_{𝐿𝐷𝐿𝐶}(𝒦). Define now 𝒫(𝐺)→−^𝛼𝒫(𝐺)=𝒫(𝐺)−−−−→^1𝒫(𝐺)𝒫(𝐺), 𝒫(𝑀)−−−→^𝛽𝑜𝑝𝒫(𝑀)=𝒫(𝑀)−→^∅𝒫(𝑀), and get an 𝐿-𝐅𝐂^𝐶-morphism 𝐻_{𝐿𝐷𝐿𝐶}(𝒦)−−−−→^(𝛼,𝛽)𝐻_{𝐿𝐷𝐿𝐶}(𝒦). However, 𝛽^𝑜𝑝∘𝐻∘𝛼(𝐺)=𝛽^𝑜𝑝∘𝐻(𝐺)=𝛽^𝑜𝑝(𝑀)=∅≠𝑀=𝐻(𝐺) and, therefore, |𝒦|−−−−→^(𝛼,𝛽)|𝒦| is not an 𝐿-𝐅𝐂^𝐷𝑀𝑅-morphism.

5.4.3. 𝐿-𝐅𝐂^𝐶 versus 𝐿-𝐅𝐂^𝐷𝑀𝑅

The previous two subsections gave the functors 𝐿-𝐅𝐂−−−−−→^𝐶_∗^{𝐻𝐿𝐶𝐿𝐷}𝐿-𝐅𝐂^𝐷𝑀𝑅, 𝐿-𝐅𝐂−−−−−→^{𝐻𝐷𝑀𝑅𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶.

Definition 98. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂_∗^𝐷𝑀𝑅 is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐷𝑀𝑅, with the same objects, whose morphisms 𝒦−→₁^𝑓𝒦₂ have the property that the maps 𝐿→−^𝐺1𝛼𝐿^𝐺2, 𝐿−−−→^{𝑀2𝛽𝑜𝑝}𝐿^𝑀1 are surjective.

It is easy to see that there exist the restrictions 𝐿-𝐅𝐂−−−−−→^𝐶_∗^{𝐻∗𝐿𝐶𝐿𝐷}𝐿-𝐅𝐂_∗^𝐷𝑀𝑅, 𝐿-𝐅𝐂−−−−−→_∗^{𝐻𝐷𝑀𝑅∗𝐿𝐷𝐿𝐶}𝐿-𝐅𝐂^𝐶_∗ of the previously mentioned two functors. A natural question could be whether the pair makes an adjoint situation. The following result, however, provides a negative answer.

Theorem 99. In general, 𝐻^∗_{𝐿𝐶𝐿𝐷} (resp., 𝐻^∗_{𝐿𝐷𝐿𝐶}) is not a left adjoint to 𝐻^∗_{𝐿𝐷𝐿𝐶} (resp., 𝐻^∗_{𝐿𝐶𝐿𝐷}).

Proof. Suppose that 𝐻^∗_{𝐿𝐶𝐿𝐷} is a left adjoint to 𝐻^∗_{𝐿𝐷𝐿𝐶}. Then every object 𝒦 of 𝐿-𝐅𝐂^𝐶_∗ has an 𝐻^∗_{𝐿𝐷𝐿𝐶}-universal map, that is, an 𝐿-𝐅𝐂^𝐶_∗-morphism 𝒦−−−−−−→^{𝜂=(𝛼,𝛽)}(𝐻^∗_{𝐿𝐷𝐿𝐶}∘𝐻^∗_{𝐿𝐶𝐿𝐷}(𝒦)=(𝐿^𝐺,𝐿^𝑀,𝐼)̂) and, therefore, 𝐼(𝑔,𝛽^𝑜𝑝(𝑡))=𝐼(̂𝛼(𝑔),𝑡) for every 𝑔∈𝐺 and every 𝑡∈𝐿^𝑀. It follows that 𝐼(𝑔,𝛽^𝑜𝑝(𝑡))∈{⟂_𝐿,⊤_𝐿}. Define 𝐺={𝑔}, 𝑀={𝑚}, 𝐿=([0,1],⋁,⋀) (the unit interval, with the standard lattice-theoretic structure) and 𝐼=1/2. There exists then a unique map 𝐿→−^𝑀!𝑀, and, moreover, one obtains that 1/2=𝐼(𝑔,!(1/2))=𝐼(̂𝛼(𝑔),1/2)∈{0,1}, which is an obvious contradiction.
Suppose that 𝐻^∗_{𝐿𝐷𝐿𝐶} is a left adjoint to 𝐻^∗_{𝐿𝐶𝐿𝐷}. Then every object 𝒦 of 𝐿-𝐅𝐂_∗^𝐷𝑀𝑅 has an 𝐻^∗_{𝐿𝐶𝐿𝐷}-universal map, that is, an 𝐿-𝐅𝐂_∗^𝐷𝑀𝑅-morphism 𝒦−−−−−−→^{𝜂=(𝛼,𝛽)}(𝐻^∗_{𝐿𝐶𝐿𝐷}∘𝐻^∗_{𝐿𝐷𝐿𝐶}(𝒦)=(𝐿^𝐺,𝐿^𝑀,𝐼)̂) and thus, 𝐼(𝑔,𝛽^𝑜𝑝(𝑡))=𝐼(̂𝛼(𝑔),𝑡) for every 𝑔∈𝐺 and every 𝑡∈𝐿^𝑀. Similar to the aforementioned, we get a contradiction.

5.5. 𝐿-𝐅𝐂^𝐶 versus 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅

The reader should notice that the results of Section 5.1 can be easily restricted to the case of the category 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅. In particular, each of Theorems 48 and 55 provides the following result.

Definition 100. Given an 𝐋-algebra 𝐿, 𝐿-𝐅𝐂^𝐶_∗ is the (nonfull) subcategory of the category 𝐿-𝐅𝐂^𝐶, with the same objects and with morphisms 𝒦−→₁^𝑓𝒦₂ such that the maps 𝐺→−₁^𝛼𝐺₂, 𝑀−−−→₂^𝛽𝑜𝑝𝑀₁ are surjective.

Theorem 101. There exists the functor 𝐿-𝐅𝐂−−−−−−→^𝐶_∗^{𝐻𝐿𝐶𝐿𝐷𝑎}𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅, which is defined by the formula 𝐻_{𝐿𝐶𝐿𝐷𝑎}(𝒦−→₁^𝑓𝒦₂)=𝒦−−−−−−−−−−→₁^{(𝛼→𝐿,(𝛽𝑜𝑝)←𝐿)}𝒦₂.

Due to the already considerable size of the paper, the respective fixed-basis restrictions of the variable-basis setting of the previous subsections are left to the reader.

6. Conclusion: Open Problems

In this paper, we provided different approaches to a possible lattice-valued generalization of Formal Concept Analysis (FCA) of Ganter and Wille [31], motivated by the recent study of Denniston et al. [29] on this topic. In particular, we have constructed several categories, whose objects are lattice-valued extensions of formal contexts of FCA and whose morphisms reflect the crisp setting of Pratt [13], the lattice-valued setting of Denniston et al. [29], and the many-valued setting of Ganter and Wille [31] themselves. In the next step, we considered many possible functors between the newly defined categories. As a consequence, we embedded each of the constructed categories into its respective counterparts. The crucial difference of this paper from the motivating one of Denniston et al. [29] can be briefly formulated as follows. (i)The underlying lattices of lattice-valued formal contexts are extensions of quantales, instead of restricted to precisely commutative quantales as in [29]. (ii)All the categories of lattice-valued formal contexts and almost all of their respective functors are made variable-basis (in the sense of [4]), instead of being fixed-basis as in [29]. (iii)Unlike [29], we consider the setting of many-valued formal contexts of Ganter and Wille [31].

In the wake of the constructed functors of this paper, we can single out the following important properties of the approach of Denniston et al. [29], which is based on the category of Galois connections of [32].

Firstly, their approach falls rather out of the standard settings of Pratt as well as Ganter and Wille. On one hand, there exists an obvious isomorphism between the subcategory 2-𝐏𝐆𝐀𝐋 of the category GAL [32] of Galois connections on crisp powersets and the crisp category 2-𝐅𝐂_𝑎^𝐷𝑀𝑅 (cf. Definitions 31(1) and 37) of this paper (essentially due to Denniston et al. [29]). On the other hand, the result of Theorem 42 shows that given a unital quantale 𝐿, the categories 𝐿-𝐏𝐆𝐀𝐋 and 𝐿-𝐅𝐂_𝑎^𝐷𝑀𝑅 are, in general, nonisomorphic. The variable-basis approach complicates the case even more. Speaking metamathematically, while the notion of formal concept of FCA does involve an order-reversing Galois connection between the respective powersets in its definition, its main essence appears to come from a binary relation on two sets, that is, the set of objects (𝐺) and the set of their respective attributes (𝑀), which says whether a given object 𝑔∈𝐺 has a given property 𝑚∈𝑀. Thus, these are not the Galois connections, but their generating relations, which seem to play the main role. Moreover, Theorem 99 of the paper states that while a particular subcategory of the category of lattice-valued formal contexts in the sense of Pratt is isomorphic to a nonfull subcategory of the category of Denniston et al., it does not provide a (co)reflective subcategory of the latter category with respect to the constructed functors (at the moment though, we do not know whether one can define different functors, which do give rise to a (co)reflective subcategory in question).

Secondly, the category of lattice-valued formal contexts, based on the category of Galois connections, tends to have the nature of a fixed-basis category. More precisely, while making the setting of Denniston et al. variable-basis in the current paper, we were unable to involve the morphisms of the underlying algebras of lattice-valued formal contexts in its definition, thereby providing a quasi-variable-basis setting.

We would like to end the section with several open problems related to the setting of this paper.

6.1. Proper Way of Building Lattice-Valued FCA

The previously mentioned discussion on the difference between the settings of binary relations and Galois connections in the lattice-valued case motivates the following open problem.

Problem 1. Is it possible to build a lattice-valued approach to FCA, which is based on order-reversing Galois connections on lattice-valued powersets, which are not generated by lattice-valued relations on their respective sets of objects and their attributes?

This problem seems especially important because we have established that using order-reversing Galois connections is more general than using relations. It may even seem that using order-reversing Galois connections with the corresponding Birkhoff operators is more robust and better than using relations. Therefore, it seems important to be able to understand how to use the Birkhoff operators when a corresponding relation may not exist. The answer may not depend on defining a relation. The answer may depend on being able to do formal concept analysis without relations.

Notice that although the notions of formal concept, protoconcept, and preconcept (cf. Definition 44) can be easily defined with the help of just a Galois connection in hand, it is their respective interpretation that poses the main problem. More precisely, what will be then in place of the crucial term “an object 𝑔 has an attribute 𝑚”? It is our opinion that such an approach can still be developed, substituting the required single lattice-valued relation 𝐺×𝑀→−^𝐼𝐿 by a pair of lattice-valued relations 𝐼₁(𝑔,𝑚)=(𝛼(𝜒^1𝐿_{𝑔}))(𝑚) and 𝐼₂(𝑔,𝑚)=(𝛽(𝜒^1𝐿_{𝑚}))(𝑔), where 𝐿⇄^𝐺𝛼_𝛽𝐿^𝑀 is a given order-reversing Galois connection, which violates the conditions of Theorem 42 (notice that the case of a commutative quantale, i.e., ⋅→_𝑙⋅=⋅→_𝑟⋅, simplifies, but does not save the situation). In view of the discussion, a particular instance of Problem 1 is as follows.

Problem 2. What is the interpretation of the relations 𝐼₁ and 𝐼₂ and what kind of lattice-valued FCA can be developed through them?

An ad hoc approach could be to read 𝐼₁(𝑔,𝑚) as the degree to which “an object 𝑔 has an attribute 𝑚” and to read 𝐼₂(𝑔,𝑚) as the degree to which “an attribute 𝑚 is shared by an object 𝑔,” that is, distinguishing between the classically indistinguishable expressions “to have an attribute’’ and “an attribute is being shared.”

6.2. Adjoint Situations between Possible Approaches to Lattice-Valued FCA

In the paper, we have constructed functorial embeddings of several categories of lattice-valued formal contexts into each other. However, we failed to construct a single adjoint situation between the categories in question, being able just to obtain a negative result in this respect (Theorem 99). In view of the discussion, our next problem could be formulated as follows.

Problem 3. Construct (if possible) an adjoint situation between the categories of lattice-valued formal contexts introduced in this paper.

Notice that the problem has a close relation to the discussion of the previous subsection. More precisely, an adjoint situation between the categories of lattice-valued formal contexts in the sense of Pratt and Denniston et al., respectively, can provide a partial answer to the problem of dispensing with binary relations in lattice-valued FCA and substituting them by order-reversing Galois connections.

6.3. Proper Morphisms of Lattice-Valued Formal Contexts

In the paper, we have dealt with both lattice-valued formal contexts and their respective morphisms. Whereas the former have already been treated in the fuzzy literature, the case of the latter is still not sufficiently clear. In particular, we have provided three possible approaches to (lattice-valued) formal context morphisms, that is, that of Pratt, that of Denniston et al., and that of Ganter and Wille. Taking apart the metamathematical discussion on the fruitfulness of any of the previously mentioned approaches, the last problem of the paper can be stated then as follows.

Problem 4. What is the most suitable (if any) way of defining the morphisms of lattice-valued FCA?

All the previously mentioned open problems will be addressed in our subsequent articles on the topic of lattice-valued FCA. An interested reader is kindly invited to share the effort.

Acknowledgments

The author is grateful to Professor W. Tholen and Professor J. Steprans, who invited him to spend the first six months of the year 2012 at the Department of Mathematics and Statistics of York University (Toronto, ON, Canada), during which period the paper was prepared. The author would also like to thank J. T. Denniston, A. Melton (Kent State University), and S. E. Rodabaugh (Youngstown State University) for going through the draft version of the paper, pointing out its various deficiencies and suggesting their adjustments. This research was supported by the ESF Project no. CZ.1.07/2.3.00/20.0051 “Algebraic Methods in Quantum Logic’’ of the Masaryk University in Brno, Czech Republic.