Research Article | Open Access
Lattice-Valued Topological Systems as a Framework for Lattice-Valued Formal Concept Analysis
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.
This paper considers a particular application of the theory of variety-based topological systems, introduced in  as a generalization of topological systems of Vickers , 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 , who considered various functorial relationships between topological systems of Vickers and lattice-valued topological spaces of Rodabaugh . Soon afterwards, the study of Guido [5, 6] followed; he used topological systems in order to construct a functor from the category of -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., ).
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 . The authors brought their theory into maturity in , where they considered its applications to both lattice-valued variable-basis topology of Rodabaugh  and -fuzzy topology of Kubiak and Šostak . Later on, they introduced a particular instance of their concept called interchange system , which was motivated by certain aspects of program semantics (the so-called predicate transformers) initiated by Dijkstra . 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  (the original notion goes back to Barr ). Moreover, the interchange system morphisms of  are precisely the Chu space morphisms (called Chu transforms) of . (In , 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 , 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  in full detail, and we brought the functor of Guido to a new level  (e.g., made it variable-basis in the sense of Rodabaugh , as well as extended the machinery of Höhle , to construct its right adjoint functor). Moreover, in , we provided a thorough categorical extension of the system spatialization procedure to the variety-based setting, the respective localification procedure being elaborated in . Additionally, certain aspects of the new theory bearing links to bitopological spaces of Kelly  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 , 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 . 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. , Denniston et al.  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 . 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 , 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 , 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 . 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 , 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.  and which is motivated by the commutative approach of fuzzy Galois connections of Bêlohlávek . 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 . 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 be a (possibly, proper or empty) class of cardinal numbers. An -algebra is a pair , which comprises a set and a family of maps (-ary primitive operations on ). An -homomorphism is a map such that the diagram (1) commutes for every . is the construct of -algebras and -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 (resp., ) be the class of -homomorphisms with injective (resp., surjective) underlying maps. A variety of -algebras are a full subcategory of , closed under the formation of products, -subobjects (subalgebras), and -quotients (homomorphic images). The objects (resp., morphisms) of a variety are called algebras (resp., homomorphisms).
Definition 3. Given a variety , a reduct of is a pair , where is a variety such that , and is a concrete functor. The pair is called then an extension of .
Definition 4. is the variety of -semilattices, that is, partially ordered sets (posets), which have arbitrary (meets and joins, resp.).
Given a -semilattice , we have both and (however, only one of them is respected by -semilattice homomorphisms) and denote its largest (resp., smallest) element (resp., ). Given a -semilattice , we denote its respective -semilattice and vice versa. The standard result of [50, Sections ] says that every -semilattice homomorphism has the upper adjoint map , which is characterized uniquely by the condition iff for every and every , which, for order-preserving maps only, is equivalent to the properties and (given an algebra , denotes the identity map on ). The explicit formula for the map is given by , whereas the map itself is easily shown to be -preserving. The dual description of the lower adjoint map (based on a -semilattice homomorphism and denoted by ) is left to the reader (see also Section 2.3 on Galois connections).
Definition 5. is the variety of quantales, that is, triples where(1) is a -semilattice; (2) is a semigroup; that is, for every ;(3) distributes across from both sides; that is, and for every and every .
Every quantale has two residuations, which are induced by its binary operation (called multiplication) and which are defined by and (notice that “" (resp., “") 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. ). These operations have the standard properties of poset adjunctions [50, Sections ] or (order-preserving) Galois connections  (see Section 2.3 of this paper), for example, iff iff . 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 and of a quantale have the following properties.(1) and for every and every . In particular, and for every . (2) and for every and every . In particular, and for every . (3) and for every . (4) and for every .
Proof. To give the flavor of the employed machinery, we show the proofs of some of the claims of the items. Ad (3). implies implies . Ad (4). Given , iff iff iff .
Moreover, iff iff iff .
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 with commutative multiplication; that is, for every , .
Definition 8. UQuant is the variety of unital quantales, which are quantales having an element such that the triple is a monoid; that is, for every .
Definition 9. SRSQuant (resp., SLSQuant) is the variety of strictly right-sided (resp., left-sided) quantales, which are quantales such that (resp., ) for every . 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 , in which are considered as primitive operations (cf. Definition 1) and which additionally are equipped with the complement operation , assigning to every an element such that and .
The varieties CBAlg, Frm, STSQuant, UQuant, and 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 of -semilattices with 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 is denoted by . However, the dual of Frm uses the already accepted notation Loc . Given a homomorphism , the corresponding morphism of the dual category is denoted by and vice versa. Every algebra of a variety gives rise to the subcategory of either or , whose only morphism is the identity map .
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 , there exists the forward (resp., backward) powerset operator (resp., ) defined by (resp., ). 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 of a variety and a set , there exists the -powerset of , namely, the set of all maps . Equipped with the pointwise algebraic structure involving that of , provides an algebra of . In the paper, we will often use specific elements of this powerset algebra. More precisely, every gives rise to the constant map with the value . Moreover, if extends the variety of -semilattices, then every and every give rise to the map (-characteristic map of ), which is defined by Of particular importance will be the characteristic maps of singletons, that is, or , the latter case assuming additionally that 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 , being different from the respective lattice-valued approaches of Rodabaugh.
Theorem 12. Given a variety , which extends , every subcategory of provides the functor , which is defined by , where .
Proof. To show that is -preserving, notice that given , for every , .
To show that the functor preserves composition, notice that given -morphisms and , for every and every , .
The case is denoted by 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  (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  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 , which extends , every subcategory of gives rise to the functor defined by , where . (2) Let be a variety, which extends , and let be a subcategory of such that for every -morphism , the map is -preserving. Then there exists the functor defined by , where .
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 - (resp., -) homomorphisms and , it follows that (resp., ).
The respective generalization of the backward powerset operator is even stronger (see, e.g., ), 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 , every subcategory of induces the functor , which is defined by , where .
Proof. To show that is an -homomorphism, notice that given and , for every , it follows that
To show that the functor preserves composition, notice that given -morphisms and , for every , .
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 or . Given a variety , which extends , every subcategory of provides the functor , which is defined by , where .
Proof. The proof is a straightforward modification of that of Theorem 14.
Theorem 16. (1)Given a variety , which extends , every subcategory of gives rise to the functor defined by , where . (2)Given a variety , which extends , every subcategory of gives rise to the functor defined by , where .(3)Let be a variety, which extends , and let be a subcategory of such that for every -morphism , the map is -preserving. Then there exists the functor defined by , where .
Theorem 17. (1) Given a variety , every subcategory of induces the functor , which is defined by , where .(2) Suppose that either or . Given a variety , which extends , every subcategory of provides the functor , which is defined by , where .
The reader should notice that the functors of Theorem 17 are not standard in the fuzzy community.
2.3. Galois Connections
Definition 18. An order-preserving (resp., -reversing) Galois connection is a tuple , where , are posets and are maps such that for every and every , iff (resp., iff ).
As a consequence, one easily gets the following well-known features of Galois connections.
Theorem 19. Every order-preserving (resp., order-reversing) Galois connection has the following properties.(1) and (resp., and ). (2) Both and are order-preserving (resp., order-reversing). (3) and (resp., and ) provided that the joins and meets in question exist in the domains. (4) and .
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 , where , are posets and are order-preserving (resp., order-reversing) maps such that and (resp., and ).
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 provides the order-preserving Galois connection (resp., ). (2) Every element of a quantale provides the order-preserving Galois connections and . (3) Every map provides the order-preserving Galois connection . (4) Given a variety extending , every -morphism provides the order-preserving Galois connection .
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 , we introduced the concept of variety-based topological systems as a generalization of Vickers’ topological systems  mentioned in the Introduction section. Moreover, in , 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 subcategory S of , and a reduct of , is the category, concrete over the product category , which comprises the following data.
Objects. Tuples (-topological systems or -systems), where is a -object, whereas is a map (-satisfaction relation on ) such that is a -homomorphism for every .
Morphisms. are all those -morphisms , , which satisfy the property for every and every (-continuity).
The following notational remark applies to all the categories of systems introduced in this paper.
Remark 23. The case is called (variety-based) fixed-basis approach, whereas all other cases are subsumed under (variety-based) variable-basis approach. Moreover, if , then the notation is truncated to , whereas if , then the notation is shortened to . In the latter case, the objects (resp., morphisms) are truncated to (resp., ).
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 , then the category , where is the two-element complete Boolean algebra, is isomorphic to the category TopSys of topological systems of Vickers . (2) If , then the category is the category of lattice-valued topological systems of Denniston et al. [8, 9]. (3) If , then the category is the category of lattice-valued interchange systems of Denniston et al. [11, 29, 30], which is also the category of lattice-valued transformer systems of Denniston et al.  (notice that both categories are fixed-basis). (4) If , then the category is the category of Chu spaces over the set of Pratt . In particular, the objects of are called formal contexts in Formal Concept Analysis of Ganter and Wille , whereas the objects of the category 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 ) illustrates the concept of a topological system morphism.
Definition 25. Letting , (notice that “GalCon’’ stands for “Galois connections’’) is the full subcategory of the category , whose objects are those -topological systems , which satisfy the following two conditions: (1); (2) is a partial order on .
The reader can easily see (or verify) that the objects of the category 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 , in the sense that while the latter category uses Galois connections as its objects, the category uses them as morphisms. It will be the topic of our future research, to consider the properties of the category (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 (up to the knowledge of the author) does not allow the previously mentioned system interpretation. Indeed, the already mentioned adjunction condition now reads as iff for every and every . With the help of the dual partial order on , we do get an order-preserving Galois connection . It is unclear though how to incorporate in the system setting another order-reversing Galois connection (or ), that is, how to define the morphism composition law of the possible category (notice that and 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 .
Definition 26. Given a variety , a subcategory of , and a reduct of , (notice that “’’ stands for “modified’’) is the category, concrete over the product category , which comprises the following data.
Objects. Tuples , where is a -object and is a map such that is a -homomorphism for every .
Morphisms. are -morphisms such that for every and every .
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 -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 . Moreover, it is easy to provide the conditions, under which the system settings of Definitions 22 and 26 coincide.
Definition 27. Given a subcategory of , (resp., ) is the nonfull subcategory of the category (resp., ), which has the same objects and whose morphisms are such that (resp., ) is an S- (resp., -) isomorphism.
Theorem 28. There exists the isomorphism , which is defined by the formula .
Proof. Straightforward computations.
As a consequence, we obtain that there is no difference between the (fixed-basis) categories and . 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 subcategory of , and a reduct of , (notice that “” stands for “alternative”) is the category, concrete over the product category , which comprises the following data.
Objects. Tuples , where is a -object, whereas is a map such that is a -homomorphism for every .
Morphisms. are such that for every and every .
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 subcategory of , and a reduct of , is the category, concrete over the product category , which comprises the following data.
Objects. Tuples , where is a and is a map such that is a -homomorphism for every .
Morphisms. are -morphisms such that for every and every .
The explicit restatement of Theorem 28 for the setting of the categories , 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 . In particular, we present three possible category-theoretic machineries, the first two of which stem from the approach of Denniston et al. , 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, , and also could be found at the beginning of Section 5 of . 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  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 be an extension of the variety Quant of quantales and let . (1) (notice that “” stands for “Chu”) is a new notation for the category , whose objects ((lattice-valued) formal contexts) are now denoted (in the notation of Definition 22, , which yields , and ), where (from German “Gegenstände”) is the set of context objects, (from German “Merkmale”) is the set of context attributes, and (from German “Inzidenzrelation”) is the context incidence relation. The respective morphisms ((lattice-valued) formal context morphisms) are now triples (in the notation of Definition 22, , and , and then, for every and every , ). The case of and is called the crisp case. (2) is a new notation for the category .
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 and .
At the moment, there is no agreement in the fuzzy community, whether to build formal contexts over L or , and, therefore, we just add the lower index “” 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 ) to lattice-valued FCA, the motivational article of Denniston et al.  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 be an extension of and let . (1)(notice that “” stands for “Ganter and Wille”) is a new notation for the category . (2) is a new notation for the category .
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  (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 , Denniston et al. have introduced another approach to lattice-valued FCA, which was based on the category GAL of order-preserving Galois connections of . 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  (called there complete commutative residuated lattices). Secondly, we provide a variable-basis approach (à la ) to the topic, instead of the fixed-based one of .
To begin with, we introduce some new notions and notations (induced by the respective ones of ), which will be used in the subsequent developments of the paper.
Definition 33. Every lattice-valued formal context has the following (lattice-valued) Birkhoff operators: (1), which is given by ,(2), which is given by .
Given a lattice-valued context , the notation for the elements of (resp., for the elements of ), 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 (recall our crispness remark from Definition 31(1)) provides the maps (1), which is given by ,(2), which is given by , 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 , is an order-reversing Galois connection.
Proof. For convenience of the reader, we provide the proof, which is different from that of .
To show that the map is order-reversing, notice that given , such that , and some , for every , it follows that (Theorem 6(2)) and, therefore, . The case of is similar.
To show that , notice that given and , it follows that , where relies on the first claim of Theorem 6(3). The proof of 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 a complete residuated lattice) and of [40, Proposition 3.9] (for 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 (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 -Galois triangles) .
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, (notice that “DMR” stands for “Denniston, Melton and Rodabaugh”) is the category, concrete over the product category , which comprises the following data.
Objects. Lattice-valued formal contexts (in the sense of Definition 31) such that is an object of S.
Morphisms. are -morphisms , which make the diagrams (4) commute.
It should be emphasized that Definition 36 is inspired by the respective one of , which, in its turn, comes from the definition of the category GAL of order-preserving Galois connections of . Moreover, two important features of the category are worth mentioning. Firstly, it never depends on the morphisms of the category , mentioned at the beginning of its definition (-objects, however, are employed), which is reflected well enough in its truncated ground category. Secondly, its underlying functor to the ground category is defined on objects as , 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 .
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 and , 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 “” (“alternative”) instead of “” (“modified”).
Definition 37. Given a variety , which extends Quant, and a subcategory of , is the category, concrete over the product category , which comprises the following data.
Objects. Lattice-valued formal contexts (in the sense of Definition 31) such that is an object of .
Morphisms. are -morphisms , 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 and .
Definition 38. (resp., ) is the nonfull subcategory of the category (resp., ), which has the same objects and whose morphisms are such that (resp., ) is an isomorphism in , that is, bijective.
Theorem 39. There exists the isomorphism , which is defined by the formula .
The reader should be aware that the fixed-basis approach (i.e., the case of ) will (in general) provide nonisomorphic categories and , since both of them are not dependant on S-morphisms. More precisely, it is possible to obtain the following result.
Theorem 40. Suppose that is not a singleton, and, additionally, its quantale multiplication is not a constant map to . Then the categories and are nonisomorphic.
Proof. It is easy to see that the context , where is the empty set and is the unique possible map, provides a terminal object in the category . The category , however, does not have a terminal object, which can be shown as follows.
Suppose the context is terminal in . Define a context , where is a nonempty set and (the constant map with value ). One gets two equal constant maps . Since there exists an -morphism , the diagrams (6) commute. It follows that the map in the right-hand diagram is constant (since has that property). Thus, if maps everything to , then every map , taking to the constant value of the map , gives another -morphism . The uniqueness of such morphism necessitates to be a singleton, and then is the empty set (since is not a singleton). As a consequence, we get that the map is constant as well (with the value ). Define now another context , where is a nonempty set and . Since the quantale multiplication of is not the constant map to , we get a nonconstant map (), and, therefore, a contradiction, since the left-hand side of the previous two diagrams factorizes then (with the help of some -morphism ) the nonconstant map through the constant map .
The case when is a singleton clearly provides the isomorphic categories and . The case when is equipped with the constant -valued quantale multiplication results in all context maps , being constant (with the value , which should additionally be preserved by the corresponding maps and (resp., )), 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 .
The difference between the categories and is similar to that between the categories and (which served as our main motivation for introducing the approach of , which was mentioned, but never studied, in ).
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 extends , then every lattice-valued context satisfies the following. (1) for every and every . (2) for every , and . (3) for every , and .
The previously mentioned items are equivalent to the following ones. (a) for every , and . (b) for every , and .
Proof. Ad (1). The item follows from the fact that , where uses Theorem 6(2).
Ad (2). , where (resp., ) uses item (4) (resp., item (1)) of Theorem 6.
Ad (3). , where (resp., ) uses item (4) (resp., item (1)) of Theorem 6.
Ad . Use (1) in the right-hand side of both and .
Ad . Put in any of or and get . Now, use in the right-hand side of both and , to get and , 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 , be sets and let be a unital quantale. For every order-reversing Galois connection , the following statements are equivalent.(1) There exists a map such that and .(2)(a) for every and every . (b) for every , and . (c) for every , and .(3)(a) for every , and . (b) for every , and .
Proof. and . It follows from Theorem 41.
. Define the required map by . Given and , it follows that (notice that for every ), whereas, given and , it follows that