Fuzzy 𝑍-Continuous Posets

Abstract

The aim of this paper is to generalize fuzzy continuous posets. The concept of fuzzy subset system on fuzzy posets is introduced; some elementary definitions such as fuzzy 𝑍-continuous posets and fuzzy 𝑍-algebraic posets are given. Furthermore, we try to find some natural classes of fuzzy 𝑍-continuous maps under which the images of such fuzzy algebraic structures can be preserved; we also think about fuzzy 𝑍-continuous closure operators in alternative ways. An extension theorem is presented for extending a fuzzy monotone map defined on the 𝑍-compact elements to a fuzzy 𝑍-continuous map defined on the whole set.

1. Introduction

The concept of continuous lattice was initiated by Scott in [1, 2] in a topological manner as a mathematical tool in computer sciences (domain theory). Although it has appeared in other fields of mathematics as well, such as general topology, real analysis, algebra, category theory, logic, this concept was defined later in purely order theoretical terms and now has become the one used in almost all references.

To introduce higher type variables into recursion equations, Wright et al. [3] introduced the notion of subset systems 𝑍 in the seventies, replacing the system of all directed subsets by other types of subsets enjoying a certain stability property under the monotone maps. Here, the authors devoted their study to the so-called 𝑍-inductive posets, which is really a generalization of 𝑍 version of algebraic posets. At the end of the paper, the authors suggested an attempt to study the generalized counterpart of continuous poset (lattice) obtained by replacing directed subsets by 𝑍-subsets, where 𝑍 is an arbitrary subset system. That undertaking was begun by Bandelt and Erné [4, 5] and independently by Novak [6]; there was subsequent research concerned by other authors [7–9].

On the other side, quantitative domain theory has been developed to supply models for concurrent systems. Now it forms a new focus on domain theory and has undergone active research. Rutten’s generalized (ultra)metric spaces [10], Flagg’s continuity spaces [11], and Wager’s Ω-categories [12] are good examples, which consist of basic frameworks of quantitative domain theory (cf. [13]).

In [13], Zhang and Fan investigated quantitative domains based on frames. From the work go, they defined a fuzzy partial order which is really a degree function on a nonempty set; afterwards, they defined and studied fuzzy dcpos and fuzzy domains. Yao and Shi [14, 15] studied fuzzy dcpos and their continuity over complete residuated lattices; Su and Li [16] discussed algebraic fuzzy dcpos and exploited their relationship with fuzzy domains. Furthermore, from the viewpoint of category, Hofmann and Waszkiewicz [17–19] dealt with quantitative domains; Stubbe [20, 21] made a study of quantitative completely distributive lattices; Lai and Zhang [22] presented a systematic investigation of completeness and directed completeness of Ω-categories.

In view of the increasing interest in quantitative domain theory and 𝑍-continuous posets. Therefore, it is natural to give a presentation of these matters in a more general framework. For this purpose, we are motivated to introduce the notion of fuzzy subset systems as a structure to study quantitative domain theory. We try to extend the theory of quantitative domain theory to a more general fuzzy order structure.

This paper is arranged as follows. In Section 2, we recall some basic materials related to fuzzy posets and fuzzy Galois connections. In Section 3, we give the definition of fuzzy subset systems, then present the notions of fuzzy 𝑍-continuous posets and fuzzy strongly 𝑍-continuous posets, and study the relationship between such algebraic structures. We also discuss the fuzzy 𝑍-continuous section-retraction pair between fuzzy 𝑍-continuous posets. In Section 4, we introduce the concept of fuzzy 𝑍-complete closure systems and associate a fuzzy 𝑍-continuous closure operator with a fuzzy 𝑍-complete closure system. We prove that each fuzzy 𝑍-complete closure system of a fuzzy 𝑍-continuous poset is fuzzy 𝑍-continuous. In Section 5, the notion of fuzzy 𝑍-algebraic posets is given, then some algebraic properties of such a structure are studied. An extension theorem based on 𝑍-compact elements is obtained. In the last section, conclusions are made.

2. Preliminaries

In this paper, we will use a complete residuated lattice as the structures of truth values. Such an algebraic structure is significant in fuzzy logic in a narrow sense [23, 24]. If no other conditions are imposed, in the sequel, 𝐿 always denotes a complete residuated lattice.

Definition 1. A complete residuated lattice is an algebraic structure (𝐿,∧,∨,∗,→,0,1) such that(1)(𝐿,∧,∨,0,1) is a complete lattice with the least element 0 and the greatest element 1;(2)(𝐿,∗,1) is a commutative monoid; that is, ∗ is commutative, associative, and 𝑎∗1=𝑎 holds for all 𝑎∈𝐿;(3)∗ and → form an adjoint pair, that is, for any 𝑎,𝑏,𝑐∈𝐿, 𝑎∗𝑏≤𝑐⇔𝑎≤𝑏→𝑐.

Proposition 2. For a complete residuated lattic 𝐿, one has(1)0∗𝑎=0 and 1→𝑎=𝑎,(2)𝑎≤𝑏⇔𝑎→𝑏=1,(3)(𝑎→𝑏)∗(𝑏→𝑐)≤𝑎→𝑐,(4)𝑎→⋀_𝑗∈𝐽𝑎_𝑗=⋀_𝑗∈𝐽(𝑎→𝑎_𝑗), and hence 𝑎→𝑏≤𝑎→𝑐 whenever 𝑏≤𝑐,(5)(⋁_𝑗∈𝐽𝑎_𝑗)→𝑐=⋀_𝑗∈𝐽(𝑎_𝑗→𝑐), and hence 𝑎→𝑐≥𝑏→𝑐 whenever 𝑎≤𝑏,(6)𝑎→(𝑏→𝑐)=𝑏→(𝑎→𝑐)=𝑎∗𝑏→𝑐,(7)𝑎∗(𝑎→𝑏)≤𝑏,(8)(𝑎→𝑏)∗(𝑐→𝑑)≤𝑎∗𝑐→𝑏∗𝑑,(9)𝑎→𝑏≤(𝑏→𝑐)→(𝑎→𝑐).

More properties about complete residuated lattices can be found in [24].

Let 𝑋 be a nonempty set. An 𝐿-subset on 𝑋 is a map from 𝑋 to 𝐿, and the family of all 𝐿-subsets on 𝑋 will be denoted by 𝐿^𝑋. All algebraic operations on 𝐿 can be extended pointwisely to the power set 𝐿^𝑋. That is, for any 𝐴,𝐵∈𝐿^𝑋,𝑥∈𝑋, we have 𝐴≤𝐵⇔𝐴(𝑥)≤𝐵(𝑥), (𝐴→𝐵)(𝑥)=𝐴(𝑥)→𝐵(𝑥) and (𝐴∗𝐵)(𝑥)=𝐴(𝑥)∗𝐵(𝑥).

Definition 3 (see [13, 25]). A fuzzy poset is a pair (𝑋,𝑒) such that 𝑋 is a non-empty set and 𝑒∶𝑋×𝑋→𝐿 is a map, called a fuzzy order, that satisfies for any 𝑥,𝑦,𝑧∈𝑋,(1)𝑒(𝑥,𝑥)=1;(2)𝑒(𝑥,𝑦)∗𝑒(𝑦,𝑧)≤𝑒(𝑥,𝑧);(3)𝑒(𝑥,𝑦)=𝑒(𝑦,𝑥)=1⇒𝑥=𝑦.

If 𝑒 is an 𝐿-partial order on 𝑋, then (𝑋,𝑒) is called an 𝐿-partial ordered set (simply, a fuzzy poset). To study fuzzy relational systems, Belohlávek [23] defined and studied an 𝐿-order over complete residuated lattices. It is shown in [14] that the previous notion is equivalent to Belohlávek’s one.

Example 4. (1) For a non-empty set 𝑋, the subsethood degree map sub(−,−)∶𝐿^𝑋×𝐿^𝑋→𝐿 is defined by for each pair (𝐴,𝐵)∈𝐿^𝑋×𝐿^𝑋, sub(𝐴,𝐵)=⋀_𝑥∈𝑋(𝐴(𝑥)→𝐵(𝑥)), then sub(−,−) is a fuzzy partial order on 𝐿^𝑋 and (𝐿^𝑋,sub) is a fuzzy poset. Especially, when 𝐴≤𝐵, we have sub(𝐴,𝐵)=⋀_𝑥∈𝑋(𝐴(𝑥)→𝐵(𝑥))=1.
(2) If 𝑀⊆𝑋, then (𝑀,𝑒_𝑀) is also a fuzzy poset (relative to the induced order from (𝑋,𝑒)), where 𝑒_𝑀 is the restriction of 𝑒 to 𝑀×𝑀.

Based on the introduction of fuzzy posets, the basic notions, such as join, meet, fuzzy closure operator and fuzzy Galois connection, can be established as an approach to generalizing the classic order theory. Here, we only recall some fundamental notions and give some basic properties needed in this paper. One can refer to [13–16, 23, 26–29] for further details.

Definition 5. In a fuzzy poset (𝑋,𝑒), an element 𝑥₀∈𝑋 is called a join (or meet) of a fuzzy subset 𝐴, in symbols 𝑥₀=⊔𝐴(or 𝑥₀=⊓𝐴) if(1)for any 𝑥∈𝑋, 𝐴(𝑥)≤𝑒(𝑥,𝑥₀)(or 𝐴(𝑥)≤𝑒(𝑥₀,𝑥)),(2)for any 𝑦∈𝑋, ⋀_𝑥∈𝑋(𝐴(𝑥)→𝑒(𝑥,𝑦))≤𝑒(𝑥₀,𝑦)(or ⋀_𝑥∈𝑋(𝐴(𝑥)→𝑒(𝑦,𝑥))≤𝑒(𝑦,𝑥₀)).

For any 𝑥∈𝑋, ↓𝑥∈𝐿^𝑋(or ↑𝑥∈𝐿^𝑋) is defined as for any 𝑦∈𝑋, ↓𝑥(𝑦)=𝑒(𝑦,𝑥)  (or ↑𝑥(𝑦)=𝑒(𝑥,𝑦)). For 𝐴∈𝐿^𝑋, ↓𝐴 is defined as ↓𝐴(𝑥)=⋁_𝑑∈𝑋𝐴(𝑑)∗𝑒(𝑥,𝑑). 𝐴∈𝐿^𝑋 is called a fuzzy upper set (or a fuzzy lower set) if for any 𝑥,𝑦∈𝑋, 𝐴(𝑥)∗𝑒(𝑥,𝑦)≤𝐴(𝑦)(or 𝐴(𝑥)∗𝑒(𝑦,𝑥)≤𝐴(𝑦)).

Proposition 6. Let (𝑋,𝑒) be a fuzzy poset. Then(1)𝑥₀=⊔𝐴 if and only if for any 𝑦∈𝑋, 𝑒(𝑥₀,𝑦)=⋀_𝑧∈𝑋(𝐴(𝑧)→𝑒(𝑧,𝑦)),(2)𝑥₀=⊓𝐴 if and only if for any 𝑦∈𝑋, 𝑒(𝑦,𝑥₀)=⋀_𝑧∈𝑋(𝐴(𝑧)→𝑒(𝑦,𝑧)),
particularly, when 𝐴=↓𝑥, one has 𝑒𝑥,𝑦=_𝑧∈𝑋𝑒()=𝑧,𝑥⟶𝑒𝑧,𝑦_𝑧∈𝑋𝑒𝑦,𝑧⟶𝑒(𝑥,𝑧). (1)

𝑥∈𝑋 is called the maximal (or minimal) element of 𝐴∈𝐿^𝑋, in symbols 𝑥=max𝐴(or 𝑥=min𝐴) if 𝐴(𝑥)=1 and for all 𝑦∈𝑋, 𝐴(𝑦)≤𝑒(𝑦,𝑥)(or 𝐴(𝑦)≤𝑒(𝑥,𝑦)). It is easy to check that if 𝐴 has a maximal (or minimal) element, then it is unique.

Definition 7. Given two fuzzy posets (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌), a map 𝑓∶𝑋→𝑌 is said to be fuzzy monotone if for any 𝑥,𝑦∈𝑋, 𝑒_𝑋(𝑥,𝑦)≤𝑒_𝑌(𝑓(𝑥),𝑓(𝑦)). Furthermore, a fuzzy monotone map 𝑓∶𝑋→𝑋 is called a projection on 𝑋 if for all 𝑥∈𝑋, 𝑓∘𝑓(𝑥)=𝑓(𝑥). We say a projection is a fuzzy closure operator if 𝑒(𝑥,𝑐(𝑥))=1 for all 𝑥∈𝑋.

Definition 8. Let (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) be two fuzzy posets, 𝑓∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) and 𝑔∶(𝑌,𝑒_𝑌)→(𝑋,𝑒_𝑋) two fuzzy monotone maps. The pair (𝑓,𝑔) is called a fuzzy Galois connection between (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) provided that forany𝑥∈𝑋,𝑦∈𝑌,𝑒_𝑌𝑦,𝑓(𝑥)=𝑒_𝑋𝑔𝑦,𝑥, (2) where 𝑓 is called the upper adjoint of 𝑔 and dually 𝑔 is the lower adjoint of 𝑓.

Obviously, a fuzzy Galois connection is an extension of a crisp Galois connection. The crisp Galois connection is defined as follows: 𝑦≤𝑓(𝑥)⇔𝑔(𝑦)≤𝑥 for any 𝑥∈𝑋, 𝑦∈𝑌, and its relative properties can be found in [30].

Example 9. Let 𝑓∶𝑋→𝑌 be a map. For each 𝐵∈𝐿^𝑌, let 𝑓^←_𝐿(𝐵)=𝐵∘𝑓. Then we obtain a powerset operator: 𝑓^←_𝐿∶𝐿^𝑌→𝐿^𝑋. Conversely, define a powerset operator 𝑓^→_𝐿∶𝐿^𝑋→𝐿^𝑌 by forany𝑦∈𝑌,𝐴∈𝐿^𝑋,𝑓^→_𝐿(𝐴)𝑦=_𝑥∈𝑋𝐴(𝑥)∗𝑒_𝑌𝑦,𝑓(𝑥). (3) Then (𝑓^→_𝐿,𝑓^←_𝐿) is a fuzzy Galois connection between (𝐿^𝑋,sub) and (𝐿^𝑌,Low).

Proposition 10. Let (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) be two fuzzy posets, 𝑓∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) and 𝑔∶(𝑌,𝑒_𝑌)→(𝑋,𝑒_𝑋) two maps. Then the following are equivalent:(1)(𝑓,𝑔) is a fuzzy Galois connection,(2)𝑓 is fuzzy monotone and 𝑔(𝑦)=min𝑓^←_𝐿(↑𝑦) for all 𝑦∈𝑌,(3)𝑔 is fuzzy monotone and 𝑓(𝑥)=max𝑓^←_𝐿(↓𝑥) for all 𝑥∈𝑋.

Proposition 11. Let (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) be two fuzzy posets, 𝑓∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) and 𝑔∶(𝑌,𝑒_𝑌)→(𝑋,𝑒_𝑋) two maps.(1)If 𝑓 is a fuzzy monotone map and has a lower adjoint, then for any 𝐴∈𝐿^𝑋 such that ⊓𝐴 exists, 𝑓(⊓𝐴)=⊓𝑓^→_𝐿(𝐴).(2)If 𝑔 is a fuzzy monotone map and has an upper adjoint, then for any 𝐷∈𝐿^𝑌 such that ⊔𝐷 exists, 𝑔(⊔𝐷)=⊔𝑔^→_𝐿(𝐷).

For any map 𝑓∶𝑋→𝑌, we denote the corestriction to the image as 𝑓^∘∶𝑋→𝑓(𝑋) and the inclusion of the image into 𝑌 accordingly as 𝑓_∘∶𝑓(𝑋)→𝑌. Thus, each 𝑓 has the decomposition 𝑓=𝑓_∘𝑓^∘. If 𝑋=𝑌, then 𝑓^∘𝑓_∘ is the restriction and corestriction 𝑓|_𝑓(𝑋)∶𝑓(𝑋)→𝑓(𝑋).

Proposition 12. Let (𝑋,𝑒) be a fuzzy poset and 𝑓∶𝑋→𝑋 a fuzzy monotone map. Then the following are equivalent:(1)𝑓 is a fuzzy closure operator,(2)(𝑓_∘,𝑓^∘) is a fuzzy Galois connection between (𝑓(𝑋),𝑒_𝑓(𝑋)) and (𝑋,𝑒_𝑋),(3)There is a fuzzy Galois connection (𝑔,𝑑) between some fuzzy posets (𝑆,𝑒_𝑆) and (𝑋,𝑒_𝑋) such that 𝑓=𝑔𝑑.

3. Fuzzy Subset Systems and Fuzzy 𝑍-Continuous Posets

In this section, we present the definition of fuzzy subset systems, and in such a framework we propose the notions of fuzzy 𝑍-continuous posets and fuzzy strongly 𝑍-continuous posets and study the relationship between them. We finally discuss the fuzzy 𝑍-continuous section-retraction pair between fuzzy 𝑍-continuous posets and get the similar results in [29].

Let FPO denote the category of all fuzzy posets with fuzzy monotone maps as morphisms and let FSET denote the category of fuzzy subsets with fuzzy maps as morphisms.

Definition 13. A fuzzy subset system on FPO is a functor 𝑍_𝐿∶FPO→FSET satisfying the following conditions.(1)For any fuzzy poset (𝑋,𝑒_𝑋), 𝑍_𝐿(𝑋)⊆𝐿^𝑋.(2)If (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) are two fuzzy posets and 𝑓∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) a fuzzy monotone map, then for any 𝐴∈𝑍_𝐿(𝑋), 𝑍_𝐿(𝑓)(𝐴)=𝑓^→_𝐿(𝐴)∈𝑍_𝐿(𝑌).(3)For any 𝑥∈𝑋, ↓𝑥∈𝑍_𝐿(𝑋).

It is clear that in order to define a fuzzy subset system 𝑍_𝐿, it suffices to define its object maps satisfying the previous conditions. Obviously, it is exactly the formal generalization of the classic definition of a 𝑍-subset system in [3].

Let (𝑋,𝑒) be a fuzzy poset. The following are some examples of fuzzy subset systems.(1)𝒟_𝐿(𝑋) is the family of all fuzzy direct subsets of 𝑋.(2)𝒫_𝐿(𝑋) is the family of all fuzzy arbitrary subsets of 𝑋.(3)ℒ_𝐿(𝑋) is the family of all fuzzy lower subsets of 𝑋.

Based on a commutative, unital quantale Ω, the authors [22, 31, 32] introduced the concept of a class of weights and studied the complete properties of such a structure from the viewpoint of category, where 𝑍_𝐿(𝑋) is a fuzzy lower set and 𝑍_𝐿(𝑓)=𝑓^→_𝐿, but 𝑍_𝐿(𝑓)(𝐴)=𝑓^→_𝐿(𝐴)∈𝑍_𝐿(𝑌) is not required.

Definition 14. A fuzzy poset (𝑋,𝑒) is said to be fuzzy 𝑍-complete if for any 𝐴∈𝑍_𝐿(𝑋), ⊔_𝑋𝐴 exists. A fuzzy subset 𝐼 of a fuzzy poset (𝑋,𝑒) is called a fuzzy 𝑍-ideal of 𝑋 if it is a fuzzy lower set generated by some 𝐴∈𝑍_𝐿(𝑋); that is, for any fuzzy 𝑍-ideal 𝐼, there exists 𝐴∈𝑍_𝐿(𝑋) with 𝐼=↓𝐴. The sets of all fuzzy 𝑍-subsets and all fuzzy 𝑍-ideals on 𝑋 are denoted by 𝑍_𝐿(𝑋) and 𝑍𝐼_𝐿(𝑋), respectively.

Definition 15. If (𝑋,𝑒) is a fuzzy 𝑍-complete poset, then for any 𝑥,𝑦∈𝑋, define ⇓_𝑍𝑥∈𝐿^𝑋 by ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒(𝑥,⊔𝐼)⟶𝐼𝑦. (4)
⇓_𝑍∶𝑋→𝐿^𝑋 is called the fuzzy 𝑍-way-below relation. For 𝑥∈𝑋, if ⇓_𝑍𝑥(𝑥)=1, then we call 𝑥 a compact element in 𝑋, and all compact elements in 𝑋 are denoted by 𝐾(𝑋).

Next we give an equivalent definition of the fuzzy 𝑍-way-below relation in terms of fuzzy 𝑍-subsets.

Proposition 16. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset. Then for any 𝑥,𝑦∈𝑋, _{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒()𝑦=𝑥,⊔𝐼⟶𝐼_{𝑆∈𝑍𝐿(𝑋)}𝑒(𝑥,⊔𝑆)⟶_𝑑∈𝑋𝑆(𝑑)∗𝑒𝑦,𝑑. (5) That is, ⇓_𝑍𝑥(𝑦)=⋀_{𝑆∈𝑍𝐿(𝑋)}(𝑒(𝑥,⊔𝑆)→(⋁_𝑑∈𝑋𝑆(𝑑)∗𝑒(𝑦,𝑑))).

Some basic properties of the fuzzy relation are listed in the following.

Proposition 17. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset. For any 𝑥,𝑦,𝑢,𝑣∈𝑋, then(1)⇓_𝑍𝑥≤↓𝑥; (2)𝑒(𝑢,𝑥)∗⇓_𝑍𝑦(𝑥)∗𝑒(𝑦,𝑣)≤⇓_𝑍𝑣(𝑢).

Proof. (1) By Definition 13(3), for any 𝑥∈𝑋, ↓𝑥∈𝑍𝐼_𝐿(𝑋), then ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒()𝑦()𝑦𝑦𝑦.𝑥,⊔𝐼⟶𝐼≤𝑒𝑥,⊔↓𝑥⟶↓𝑥=1⟶↓𝑥=↓𝑥 (6)
(2) Is straightforward.

Definition 18. A fuzzy 𝑍-complete poset (𝑋,𝑒) is called a fuzzy 𝑍-continuous poset if for any 𝑥∈𝑋, ⇓_𝑍𝑥∈𝑍_𝐿(𝑋)(or ⇓_𝑍𝑥∈𝑍𝐼_𝐿(𝑋)) and 𝑥=⊔⇓_𝑍𝑥. A fuzzy 𝑍-continuous poset is called strongly fuzzy 𝑍-continuous if the fuzzy 𝑍-way-below relation has the interpolation property: for any 𝑥,𝑦∈𝑋, ⇓_𝑍𝑥(𝑦)=⋁_𝑑∈𝑋⇓_𝑍𝑥(𝑑)∗⇓_𝑍𝑑(𝑦).

Fuzzy 𝒟-continuous posets are known in the literature as fuzzy domains. See [11–15, 18, 19, 25, 29] for further details. Fuzzy 𝒫-continuous posets are known as fuzzy completely distributive lattices which have been widely studied by [20–22] from the viewpoint of category.

Definition 19. A fuzzy subset system 𝑍_𝐿 is said to be fuzzy union-complete if for any Φ∈𝑍_𝐿(𝑍_𝐿(𝑋)), ⊔Φ exists and ⊔Φ∈𝑍_𝐿(𝑋).

Remark 20. The fuzzy subset systems 𝒟_𝐿, 𝒫_𝐿, and ℒ_𝐿 are fuzzy union-complete.

Proof. It is trivial that the statement holds for 𝒫_𝐿 and ℒ_𝐿. We now give the proof in terms of 𝒟_𝐿. Recall that a fuzzy subset 𝐷 is a fuzzy directed subset if ⋁_𝑥∈𝑋𝐷(𝑥)=1, and for any 𝑎,𝑏∈𝑋, 𝐷(𝑎)∗𝐷(𝑏)≤⋁_𝑑∈𝑋𝐷(𝑑)∗𝑒(𝑎,𝑑)∗𝑒(𝑏,𝑑).
For any Φ∈𝒟_𝐿(𝒟_𝐿(𝑋)), put 𝜙=⋁_{𝜑∈𝒟𝐿(𝑋)}Φ(𝜑)∗𝜑. We now show that ⊔Φ=𝜙 and 𝜙∈𝒟_𝐿(𝑋). For each 𝜓∈𝒟_𝐿(𝑋), sub𝜙,𝜓=_𝑥∈𝑋𝜙(𝑥)(𝑥)=⟶𝜓_{𝑥∈𝑋𝜑∈𝒟𝐿(𝑋)}Φ𝜑(𝑥)(𝑥=∗𝜑⟶𝜓)_{𝑥∈𝑋𝜑∈𝒟𝐿(𝑋)}Φ𝜑⟶𝜑(𝑥)(𝑥)=⟶𝜓_{𝜑∈𝒟𝐿(𝑋)}Φ𝜑⟶_𝑥∈𝑋𝜑(𝑥)(𝑥)=⟶𝜓_{𝜑∈𝒟𝐿(𝑋)}Φ𝜑⟶sub𝜑,𝜓_𝑥∈𝑋,𝜙(𝑥)=_{𝑥∈𝑋𝜑∈𝒟𝐿(𝑋)}Φ𝜑∗𝜑(𝑥)=_{𝜑∈𝒟𝐿(𝑋)}Φ𝜑∗_𝑥∈𝑋𝜑(𝑥)=1, (7) and for any 𝑎,𝑏∈𝑋, 𝜙(𝑎)∗𝜙(𝑏)=_{𝜑1,𝜑2∈𝒟𝐿(𝑋)}Φ𝜑₁∗𝜑₁(𝑎)∗Φ𝜑₂∗𝜑₂(𝑏)≤_{𝜑1,𝜑2∈𝒟𝐿(𝑋)𝜑∈𝒟𝐿(𝑋)}Φ𝜑∗sub𝜑₁∗,𝜑sub𝜑₂,𝜑∗𝜑₁(𝑎)∗𝜑₂(𝑏)≤_{𝜑∈𝒟𝐿(𝑋)}Φ𝜑(𝑎)(𝑏)≤∗𝜑∗𝜑_{𝑑∈𝑋𝜑∈𝒟𝐿(𝑋)}Φ𝜑(𝑑)()()=∗𝜑∗𝑒𝑎,𝑑∗𝑒𝑏,𝑑_𝑑∈𝑋𝜙(𝑑)∗𝑒(𝑎,𝑑)∗𝑒(𝑏,𝑑). (8)

Lemma 21. In a fuzzy union-complete subset system 𝑍_𝐿, if (𝑋,𝑒) is a fuzzy 𝑍-continuous poset, then for any 𝐴∈𝑍_𝐿(𝑋), we have ⋁_𝑥∈𝑋𝐴(𝑥)∗⇓_𝑍𝑥∈𝑍_𝐿(𝑋).

Proof. Since (𝑋,𝑒) is fuzzy 𝑍-continuous, then ⇓_𝑍𝑥∈𝑍_𝐿(𝑋) for all 𝑥∈𝑋. Hence the map 𝑓=⇓_𝑍∶𝑋→𝑍_𝐿(𝑋) is welldefined. It is easy to verify that ⇓_𝑍 is fuzzy monotone, then by Definition 13(2), for any 𝐴∈𝑍_𝐿(𝑋), we have 𝑓^→_𝐿(𝐴)=⋁_𝑥∈𝑋𝐴(𝑥)∗sub(−,⇓_𝑍𝑥)∈𝑍_𝐿(𝑍_𝐿(𝑋)). Since 𝑍_𝐿 is fuzzy union-complete, to show ⋁_𝑥∈𝑋𝐴(𝑥)∗⇓_𝑍𝑥∈𝑍_𝐿(𝑋), it suffices to show that ⊔(⋁_𝑥∈𝑋𝐴(𝑥)∗sub(−,⇓_𝑍𝑥))=⋁_𝑥∈𝑋𝐴(𝑥)∗⇓_𝑍𝑥. To this end, for any 𝜓∈𝑍_𝐿(𝑋), sub_𝑥∈𝑋𝐴(𝑥)∗⇓_𝑍=𝑥,𝜓_{𝑦∈𝑋𝑥∈𝑋}𝐴(𝑥)∗⇓_𝑍𝑥𝑦𝑦=⟶𝜓_𝑥∈𝑋𝐴(𝑥)⟶_𝑦∈𝑋⇓_𝑍𝑥𝑦𝑦=⟶𝜓_𝑥∈𝑋𝐴(𝑥)⟶sub⇓_𝑍=𝑥,𝜓_𝑥∈𝑋𝐴(𝑥)⟶_{𝜑∈𝑍𝐿(𝑋)}sub𝜑,⇓_𝑍𝑥⟶sub=𝜑,𝜓_{𝜑∈𝑍𝐿(𝑋)𝑥∈𝑋}𝐴(𝑥)∗sub𝜑,⇓_𝑍𝑥⟶sub=𝜑,𝜓_{𝜑∈𝑍𝐿𝑋}𝑓^→_𝐿(𝐴)𝜑⟶sub𝜑,𝜓, (9) which indicates that ⊔𝑓^→_𝐿(𝐴)=⋁_𝑥∈𝑋𝐴(𝑥)∗⇓_𝑍𝑥.

Lemma 22. In a fuzzy union-complete subset system 𝑍_𝐿, if (𝑋,𝑒) is a fuzzy 𝑍-continuous poset, and for any 𝐴∈𝑍_𝐿(𝑋), set 𝑎=⊔𝐴, then ⇓_𝑍𝑎=⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑 holds.

Proof. We first show that ⊔(⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑)=⊔𝐴. For any 𝑦∈𝑋, _{𝑥∈𝑋𝑑∈𝑋}𝐴(𝑑)∗⇓_𝑍𝑑(𝑥=)⟶𝑒𝑥,𝑦_{𝑥∈𝑋𝑑∈𝑋}𝐴(𝑑)⟶⇓_𝑍𝑑(𝑥)=⟶𝑒𝑥,𝑦_𝑑∈𝑋𝐴(𝑑)⟶_𝑥∈𝑋⇓_𝑍𝑑(𝑥)=⟶𝑒𝑥,𝑦_𝑑∈𝑋𝐴(𝑑)⟶𝑒⊔⇓_𝑍=𝑑,𝑦_𝑑∈𝑋𝐴(𝑑).⟶𝑒𝑑,𝑦=𝑒⊔𝐴,𝑦 (10)
On the one hand, for any 𝑧∈𝑋, by Proposition 17(2), _𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑧)≤_𝑑∈𝑋𝑒(𝑑,⊔𝐴)∗⇓_𝑍𝑑(𝑧)≤_𝑑∈𝑋⇓_𝑍𝑎(𝑧)=⇓_𝑍𝑎(𝑧). (11)
On the other hand, since the fuzzy subset system is fuzzy union-complete, by Lemma 21, we have ⇓_𝑍𝑎(𝑧)=_{𝐼∈𝑍𝐼𝐿(𝑋)}(𝑒(𝑎,⊔𝐼)⟶𝐼(𝑧))≤𝑒𝑎,⊔_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑧)=𝑒(𝑎,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑧)=_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑧). (12) Therefore, ⇓_𝑍𝑎=⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑.

Proposition 23. Let a fuzzy subset system 𝑍_𝐿 be fuzzy union-complete. If (𝑋,𝑒) is a fuzzy 𝑍-continuous poset, then for any 𝑥,𝑦∈𝑋, ⇓_𝑍𝑥𝑦=_{𝐴∈𝑍𝐿(𝑋)}𝑒(𝑥,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦. (13)

Proof. For any 𝐴∈𝑍_𝐿(𝑋), ⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑∈𝑍𝐼_𝐿(𝑋), then by Lemma 22,⇓_𝑍𝑥𝑦∗𝑒(𝑥,⊔𝐴)=⇓_𝑍𝑥𝑦∗𝑒𝑥,⊔_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑=𝑒𝑥,⊔_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑∗_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒()𝑦≤𝑥,⊔𝐼⟶𝐼_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦=_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦. (14) This implies that ⇓_𝑍𝑥(𝑦)≤𝑒(𝑥,⊔𝐴)→(⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑦)).
By the arbitrariness of 𝐴, we have ⇓_𝑍𝑥𝑦≤_{𝐴∈𝑍𝐿(𝑋)}𝑒(𝑥,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦. (15)
Conversely, by Proposition 17(1), 𝑒(𝑥,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦()⟶≤𝑒𝑥,⊔𝐴_𝑑∈𝑋𝐴(𝑑)∗↓𝑑𝑦. (16) Thus _{𝐴∈𝑍𝐿(𝑋)}𝑒(𝑥,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦≤⇓_𝑍𝑥𝑦. (17) Hence, ⇓_𝑍𝑥(𝑦)=⋀_{𝐴∈𝑍𝐿(𝑋)}(𝑒(𝑥,⊔𝐴)⟶(⋁_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑(𝑦))).

Theorem 24. In a fuzzy union-complete subset system 𝑍_𝐿, a fuzzy 𝑍-continuous poset (𝑋,𝑒) is exactly a strongly fuzzy 𝑍-continuous poset.

Proof. By Definition 18, it suffices to show ⇓_𝑍𝑥(𝑦)=⋁_𝑑∈𝑋⇓_𝑍𝑥(𝑑)∗⇓_𝑍𝑑(𝑦) for all 𝑥,𝑦∈𝑋. Proposition 17 implies that the left side of the equation is larger than or equal to the right side. We only need to show the other side. By Proposition 23, ⇓_𝑍𝑥𝑦=_{𝐴∈𝑍𝐿(𝑋)}𝑒(𝑥,⊔𝐴)⟶_𝑑∈𝑋𝐴(𝑑)∗⇓_𝑍𝑑𝑦≤𝑒𝑥,⊔⇓_𝑍𝑥⟶_𝑑∈𝑋⇓_𝑍𝑥(𝑑)∗⇓_𝑍𝑑𝑦=1⟶_𝑑∈𝑋⇓_𝑍𝑥(𝑑)∗⇓_𝑍𝑑𝑦=_𝑑∈𝑋⇓_𝑍𝑥(𝑑)∗⇓_𝑍𝑑𝑦. (18)

Stubbe [20] presented an elegant characterization of the fuzzy completely distributive lattices by fuzzy Galois connection; Albert and Kelly [31] and Lai and Zhang [22] gave an equivalent characterization of Φ-cocomplete Ω-category. In the next theorem, we will give a more explicit one and find construction methods which allow us to obtain a multitude of fuzzy 𝑍-continuous posets by using the given ones as building blocks.

Theorem 25. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset. Then the following statements are equivalent:(1)(𝑋,𝑒) is a fuzzy 𝑍-continuous poset;(2)for each 𝑥∈𝑋, ⇓_𝑍𝑥 is the smallest fuzzy 𝑍-ideal 𝐼 with 𝑒(𝑥,⊔𝐼)=1;(3)for each 𝑥∈𝑋, there exists a smallest fuzzy 𝑍-ideal 𝐼 with 𝑒(𝑥,⊔𝐼)=1;(4)the sup map 𝑟=(𝐼→⊔𝐼)∶𝑍𝐼_𝐿(𝑋)→𝑋 has a lower adjoint. These conditions imply(5)the sup map 𝑟=(𝐼→⊔𝐼)∶𝑍𝐼_𝐿(𝑋)→𝑋 preserves all existing inf.

Proof. (1)⇒(2): Condition (1) holds if and only if for each 𝑥∈𝑋, ⇓_𝑍𝑥∈𝑍𝐼_𝐿(𝑋) and 𝑥=⊔⇓_𝑍𝑥. Hence ⇓_𝑍𝑥 is a fuzzy 𝑍-ideal with 𝑒(𝑥,⊔⇓_𝑍𝑥)=𝑒(𝑥,𝑥)=1.
Moreover, for all 𝑦∈𝑋, 𝐼^′∈𝑍𝐼_𝐿(𝑋) with 𝑒(𝑥,⊔𝐼^′)=1, we have ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒(𝑥,⊔𝐼)⟶𝐼𝑦≤𝑒𝑥,⊔𝐼^′⟶𝐼^′𝑦=1⟶𝐼^′𝑦=𝐼^′𝑦. (19) This establishes (2).
Condition (2) trivially implies (3).
(3)⇔(4): It is easy to check that 𝑟 is a fuzzy monotone map. By Proposition 10, 𝑟 has a lower adjoint if and only if min𝑟^←_𝐿(↑𝑥) exists for all 𝑥∈𝑋, where min𝑟^←_𝐿(↑𝑥)=min{𝐼∈𝑍𝐼_𝐿(𝑋),𝑒(𝑥,⊔𝐼)=1}. This means min𝑟^←_𝐿(↑𝑥) is precisely the smallest fuzzy 𝑍-ideal 𝐼 with 𝑒(𝑥,⊔𝐼)=1.
(4)⇒(1): Let 𝑠∶𝑋→𝑍𝐼_𝐿(𝑋) be the lower adjoint of 𝑟. To show that (𝑋,𝑒) is a fuzzy 𝑍-continuous poset, it suffices to prove 𝑠(𝑥)=⇓_𝑍𝑥 for all 𝑥∈𝑋.
On the one hand, for all 𝑦∈𝑋, since 𝑠(𝑥)∈𝑍𝐼_𝐿(𝑋), then ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒()𝑦((𝑥(𝑥)𝑦(𝑥)𝑦.𝑥,⊔𝐼⟶𝐼≤𝑒𝑥,⊔𝑠))⟶𝑠=𝑠 (20)
On the other hand, 𝑠(𝑥)(𝑦)∗sub(𝑠(𝑥),𝐼)≤𝐼(𝑦), 𝑠(𝑥)𝑦≤_{𝐼∈𝑍𝐼𝐿(𝑋)}sub(𝑠(𝑥))𝑦=,𝐼⟶𝐼_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒(𝑥,⊔𝐼)⟶𝐼𝑦=⇓_𝑍𝑥𝑦. (21)
(4)⇒(5): It is immediate from Proposition 11.

In [29], the authors studied section-retraction pair of fuzzy domains in a systematic way; here, we further give an application in fuzzy 𝑍-continuous posets.

Definition 26 (see [29]). Let (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) be two fuzzy posets, 𝑠∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) and 𝑟∶(𝑌,𝑒_𝑌)→(𝑋,𝑒_𝑋) two fuzzy monotone maps. The pair (𝑠,𝑟) is called a fuzzy monotone section-retraction pair if 𝑟∘𝑠=𝑖𝑑_𝑋. In this situation, we call (𝑋,𝑒_𝑋) a fuzzy monotone retraction of (𝑌,𝑒_𝑌).

It is clear that 𝑟 is surjective and 𝑠 is injective in a fuzzy monotone section-retraction pair (𝑠,𝑟).

Definition 27. Given two fuzzy 𝑍-complete posets (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌), a fuzzy monotone map 𝑓∶(𝑋,𝑒_𝑋)→(𝑌,𝑒_𝑌) is said to be fuzzy 𝑍-continuous if for any 𝐴∈𝑍_𝐿(𝑋), 𝑓(⊔_𝑋𝐴)=⊔_𝑌𝑓^→_𝐿(𝐴).

Definition 28. A fuzzy monotone section-retraction pair (𝑠,𝑟) is called a fuzzy 𝑍-continuous section-retraction pair if (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌) are fuzzy 𝑍-complete posets and 𝑠,𝑟 are both fuzzy 𝑍-continuous, then we speak of (𝑋,𝑒_𝑋) as a fuzzy 𝑍-continuous retraction of (𝑌,𝑒_𝑌).

Lemma 29. Let the pair (𝑠,𝑟) be a fuzzy 𝑍-continuous section-retraction-pair between (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌). Then for any 𝑥∈𝑋, 𝑦∈𝑌, ⇓_𝑍𝑠(𝑥)(𝑦)≤⇓_𝑍𝑥(𝑟(𝑦)).

Proof. Note that for any 𝐼^′∈𝑍𝐼_𝐿(𝑋), 𝑠^→_𝐿(𝐼^′)∈𝑍𝐼_𝐿(𝑌), then ⇓_𝑍𝑠(𝑥)𝑦=_{𝐼∈𝑍𝐼𝐿(𝑌)}𝑒_𝑌(𝑠(𝑥))𝑦≤,⊔𝐼⟶𝐼_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑌𝑠(𝑥),⊔𝑠^→_𝐿𝐼^′⟶𝑠^→_𝐿𝐼^′𝑦=_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑌𝑠(𝑥),𝑠⊔𝐼^′⟶𝑠^→_𝐿𝐼^′𝑦≤_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑋𝑥,⊔𝐼^′⟶𝑠^→_𝐿𝐼^′𝑦=_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑋𝑥,⊔𝐼^′⟶_𝑎∈𝑋𝐼^′(𝑎)∗𝑒_𝑌(𝑎)≤𝑦,𝑠_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑋𝑥,⊔𝐼^′⟶_𝑎∈𝑋𝐼^′(𝑎)∗𝑒_𝑋𝑟𝑦(𝑠(𝑎=,𝑟))_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑋𝑥,⊔𝐼^′⟶_𝑎∈𝑋𝐼^′(𝑎)∗𝑒_𝑋𝑟𝑦=,𝑎_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑒_𝑋𝑥,⊔𝐼^′⟶𝐼^′𝑟𝑦=⇓_𝑍𝑥𝑟𝑦. (22)

Theorem 30. A fuzzy 𝑍-continuous retraction of a fuzzy 𝑍-continuous poset is also a fuzzy 𝑍-continuous poset.

Proof. Assume that (𝑠,𝑟) is a fuzzy 𝑍-continuous section-retraction-pair between (𝑋,𝑒_𝑋) and (𝑌,𝑒_𝑌), where (𝑌,𝑒_𝑌) is a fuzzy 𝑍-continuous poset, we need to show that (𝑋,𝑒_𝑋) is also a fuzzy 𝑍-continuous poset.
Since 𝑟∘𝑠=𝑖𝑑_𝑋, then for any 𝑥∈𝑋,  𝑥=𝑟(𝑠(𝑥)). Note that 𝑟 is fuzzy 𝑍-continuous and (𝑌,𝑒_𝑌) is a fuzzy 𝑍-continuous poset, then 𝑟^→_𝐿(⇓_𝑍𝑠(𝑥))∈𝑍𝐼_𝐿(𝑋) and ⊔𝑟^→_𝐿⇓_𝑍𝑠(𝑥)=𝑟⊔⇓_𝑍𝑠(𝑥)=𝑟(𝑠(𝑥))=𝑥. (23) Therefore, to prove (𝑋,𝑒_𝑋) is fuzzy 𝑍-continuous, it suffices to show that 𝑟^→_𝐿(⇓_𝑍𝑠(𝑥))=⇓_𝑍𝑥. On the one hand, for any 𝑦∈𝑋 and by Lemma 29, 𝑟^→_𝐿⇓_𝑍𝑠(𝑥)𝑦=_𝑧∈𝑌⇓_𝑍𝑠(𝑥)(𝑧)∗𝑒_𝑋(𝑧)≤𝑦,𝑟_𝑧∈𝑌⇓_𝑍𝑥(𝑟(𝑧))∗𝑒_𝑋(𝑧)≤𝑦,𝑟_𝑧∈𝑌⇓_𝑍𝑥𝑦=⇓_𝑍𝑥𝑦. (24) On the other hand, ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒()𝑦𝑟𝑥,⊔𝐼⟶𝐼≤𝑒𝑥,⊔^→_𝐿⇓_𝑍𝑠(𝑥)⟶𝑟^→_𝐿⇓_𝑍𝑠(𝑥)𝑦=𝑟^→_𝐿⇓_𝑍𝑠(𝑥)𝑦. (25) Therefore, 𝑟^→_𝐿(⇓_𝑍𝑠(𝑥))=⇓_𝑍𝑥.

4. Fuzzy 𝑍-Complete Closure Systems

In [33], the authors studied fuzzy closure systems on 𝐿-order sets, where the 𝐿-order sets are really fuzzy posets, and discussed their relationship with fuzzy closure operators. In this section, we further introduce the concept of fuzzy 𝑍-complete closure systems and associate a fuzzy 𝑍-continuous closure operator with a fuzzy 𝑍-complete closure system. We prove that each fuzzy 𝑍-complete closure system of a fuzzy 𝑍-continuous poset is fuzzy 𝑍-continuous.

Definition 31 (see [33]). If (𝑋,𝑒) is a fuzzy poset, a fuzzy closure system on 𝑋 is a subset 𝑀 of 𝑋 such that for each 𝑥∈𝑋, min{↑𝑥∗1_𝑀} exists, where 1_𝑀 is the constant fuzzy set with the value 1 on 𝑀.

Theorem 32 (see [33]). Let (𝑋,𝑒) be a fuzzy poset, 𝑐∶𝑋→𝑋 a fuzzy monotone map, and 𝑐^∘∶𝑋→𝑐(𝑋) the corestriction to the image. Then one has the following.(1)If 𝑀 is a fuzzy closure system on 𝑋 and for each 𝑥∈𝑋, 𝑐^∘(𝑥)=min{↑𝑥∗1_𝑀}, then the map 𝑐∶𝑋→𝑋 is a fuzzy closure operator.(2)If 𝑐∶𝑋→𝑋 is a fuzzy closure operator, then 𝑐(𝑋) is a fuzzy closure system on 𝑋.(3)The map defined by (1) from the set of fuzzy closure systems on 𝑋 to the set of fuzzy closure operators on 𝑋 is bijective, and its converse is the map defined by (2).

Lemma 33. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset and 𝑀⊆𝑋. Then for any 𝐴∈𝑍_𝐿(𝑀), ⊔_𝑋𝐴=⊔_𝑋𝑖^→_𝑀𝐿(𝐴). Here, 𝑖_𝑀 is an inclusion map from 𝑀 to 𝑋. Moreover, if ⊔_𝑋𝐴∈𝑀, then ⊔_𝑋𝐴=⊔_𝑀𝐴.

Proof. It is trivial that 𝑖_𝑀 is fuzzy monotone, then for any 𝐴∈𝑍_𝐿(𝑀), 𝑖^→_𝑀𝐿(𝐴)∈𝑍_𝐿(𝑋) and ⊔_𝑋𝑖^→_𝑀𝐿(𝐴) exists. For any 𝑧∈𝑋, 𝑒_𝑋⊔_𝑋𝑖^→_𝑀𝐿(𝐴)=,𝑧_𝑥∈𝑋𝑖^→_𝑀𝐿(𝐴)(𝑥)⟶𝑒_𝑋()=𝑥,𝑧_{𝑥∈𝑋𝑦∈𝑀}𝐴𝑦∗𝑒_𝑋𝑥,𝑖_𝑀𝑦⟶𝑒_𝑋()=𝑥,𝑧_𝑦∈𝑀𝐴𝑦⟶_𝑥∈𝑋𝑒_𝑋𝑥,𝑦⟶𝑒_𝑋()=𝑥,𝑧_𝑦∈𝑀𝐴𝑦⟶𝑒_𝑋𝑦,𝑧. (26) Thus ⊔_𝑋𝐴=⊔_𝑋𝑖^→_𝑀𝐿(𝐴).
Furthermore, if ⊔_𝑋𝐴∈𝑀, then for any 𝑧^′∈𝑀, we have 𝑒_𝑋⊔_𝑋𝑖^→_𝑀𝐿(𝐴),𝑧^′=𝑒_𝑀⊔_𝑋𝑖^→_𝑀𝐿(𝐴),𝑧^′=_𝑦∈𝑀𝐴𝑦⟶𝑒_𝑀𝑦,𝑧^′. (27) This shows that ⊔_𝑀𝐴=⊔_𝑋𝑖^→_𝑀𝐿(𝐴). Hence ⊔_𝑀𝐴=⊔_𝑋𝐴.

Lemma 34. Let (𝑋,𝑒) be a fuzzy poset, 𝑓∶𝑋→𝑋 a projection, and 𝑖_𝑓(𝑋)∶𝑓(𝑋)→𝑋 an inclusion map. Then (1) for any 𝐴∈𝑍_𝐿(𝑓(𝑋)), 𝑓^→_𝐿(𝑖^→_{𝑓(𝑋)𝐿}(𝐴))=𝑓^→_𝐿(𝐴); (2) for any 𝐵∈𝑍_𝐿(𝑋), 𝑓^→_𝐿(𝑓^→_𝐿(𝐵))=𝑓^→_𝐿(𝐵).

Proof. (1) For any 𝑦∈𝑓(𝑋), note that 𝑦=𝑓(𝑦), 𝑓^→_𝐿𝑖^→_{𝑓(𝑋)𝐿}(𝐴𝑦=_𝑧∈𝑋)𝑖^→_{𝑓(𝑋)𝐿}(𝐴)(𝑧)∗𝑒_𝑓(𝑋)(𝑧)=𝑦,𝑓_{𝑧∈𝑋(𝑋)𝑥∈𝑓}𝐴(𝑥)∗𝑒_𝑋(𝑧,𝑥)∗𝑒_𝑓(𝑋)(𝑧)≤𝑦,𝑓_{𝑧∈𝑋𝑥∈𝑓(𝑋)}𝐴(𝑥)∗𝑒_𝑓(𝑋)𝑓(𝑧),𝑓(𝑥)∗𝑒_𝑓(𝑋)(𝑧)≤𝑦,𝑓_{𝑧∈𝑋(𝑋)𝑥∈𝑓}𝐴(𝑥)∗𝑒_𝑓(𝑋)(𝑥)=𝑦,𝑓_{(𝑋)𝑥∈𝑓}𝐴(𝑥)∗𝑒_𝑓(𝑋)𝑦,𝑓(𝑥)=𝑓^→_𝐿(𝐴)𝑦. (28)
For the converse, 𝑓^→_𝐿𝑖^→_{𝑓(𝑋)𝐿}(𝐴)𝑦_{𝑧∈𝑋(𝑋)𝑥∈𝑓}=𝐴(𝑥)∗𝑒_𝑋(𝑧,𝑥)∗𝑒_𝑓(𝑋)(𝑧)≥𝑦,𝑓_{(𝑋)𝑥∈𝑓}𝐴(𝑥)∗𝑒_𝑓(𝑋)𝑦,𝑓(𝑥)=𝑓^→_𝐿(𝐴)𝑦. (29)
Similarly, we can prove (2).

Proposition 35. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset and 𝑓∶𝑋→𝑋 a fuzzy 𝑍-continuous projection. Then (𝑓(𝑋),𝑒_𝑓(𝑋)) is a fuzzy 𝑍-complete poset.

Proof. By Lemma 33, it suffices to show that ⊔_𝑋𝐴∈𝑓(𝑋) for each 𝐴∈𝑍_𝐿(𝑓(𝑋)). Actually, assume that 𝑎=⊔_𝑋𝐴, which means for all 𝑧∈𝑋, 𝑒_𝑋(𝑎,𝑧)=⋀_{𝑥∈𝑓(𝑋)}(𝐴(𝑥)→𝑒_𝑋(𝑥,𝑧)). Then _{(𝑋)𝑥∈𝑓}𝑓^→_𝐿(𝐴)(𝑥)⟶𝑒_𝑋()=𝑥,𝑧_{(𝑋)𝑥∈𝑓(𝑋)𝑦∈𝑓}𝐴𝑦∗𝑒_𝑓(𝑋)𝑥,𝑓𝑦⟶𝑒_𝑋()=𝑥,𝑧_{(𝑋)𝑦∈𝑓}𝐴𝑦⟶_{(𝑋)𝑥∈𝑓}𝑒_𝑓(𝑋)𝑥,𝑓𝑦⟶𝑒_𝑋()=𝑥,𝑧_{(𝑋)𝑦∈𝑓}𝐴𝑦⟶𝑒_𝑋𝑦,𝑧, (30) which implies that 𝑒_𝑋(𝑎,𝑧)=⋀_{𝑥∈𝑓(𝑋)}(𝑓^→_𝐿(𝐴)(𝑥)→𝑒_𝑋(𝑥,𝑧)).
Since 𝑓 is fuzzy 𝑍-continuous and by Lemmas 33, 34, we have 𝑎=⊔_𝑋𝑓^→_𝐿(𝐴)=⊔_𝑋𝑓^→_𝐿𝑖^→_{𝑓(𝑋)𝐿}(𝐴)=𝑓⊔_𝑋𝑖^→_{𝑓(𝑋)𝐿}(𝐴)=𝑓⊔_𝑋𝐴=𝑓(𝑎). (31)

Remark 36. The preceding proposition states that for any fuzzy 𝑍-continuous projection 𝑓∶𝑋→𝑋, the inclusion map from 𝑓(𝑋) to 𝑋 is fuzzy 𝑍-continuous.

Proof. Since for any 𝐴∈𝑍_𝐿(𝑓(𝑋)), 𝑖_𝑓(𝑋)(⊔_𝑓(𝑋)𝐴)=⊔_𝑓(𝑋)𝐴 and ⊔_𝑋𝑖^→_{𝑓(𝑋)𝐿}(𝐴)=⊔_𝑋𝐴, then by Lemma 33 and Proposition 35, we have 𝑖_𝑓(𝑋)(⊔_𝑓(𝑋)𝐴)=⊔_𝑋𝑖^→_{𝑓(𝑋)𝐿}(𝐴); that is, the inclusion map from 𝑓(𝑋) to 𝑋 is fuzzy 𝑍-continuous.

Proposition 37. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset, 𝑓∶𝑋→𝑋 a fuzzy 𝑍-continuous projection and 𝑌=𝑓(𝑋). Then for all 𝐴∈𝑍_𝐿(𝑋), 𝑓(⊔_𝑋𝐴)=⊔_𝑌𝑓^→_𝐿(𝐴)  (in other words, the corestriction of 𝑓 to 𝑌 is fuzzy 𝑍-continuous).

Proof. Since 𝑓 is fuzzy 𝑍-continuous, then for any 𝐴∈𝑍_𝐿(𝑋), 𝑓(⊔_𝑋𝐴)=⊔_𝑋𝑓^→_𝐿(𝐴). Moreover, 𝑓 is fuzzy monotone, which indicates that 𝑓^→_𝐿(𝐴)∈𝑍_𝐿(𝑌). By Lemma 33 and Proposition 35, ⊔_𝑋𝑓^→_𝐿(𝐴)=⊔_𝑌𝑓^→_𝐿(𝐴). Thus 𝑓(⊔_𝑋𝐴)=⊔_𝑌𝑓^→_𝐿(𝐴).

Theorem 38. Let (𝑋,𝑒) be a fuzzy 𝑍-continuous poset and 𝑓∶𝑋→𝑋 a fuzzy 𝑍-continuous projection. Then (𝑓(𝑋),𝑒_𝑓(𝑋)) is fuzzy 𝑍-continuous (relative to the induced fuzzy order).

Proof. By Proposition 35, (𝑓(𝑋),𝑒_𝑓(𝑋)) is a fuzzy 𝑍-complete lattice. For any 𝑥∈𝑋, 𝑓(𝑥)=𝑓∘𝑓(𝑥)=𝑓(𝑖_𝑓(𝑋)(𝑓(𝑥)))=(𝑓∘𝑖_𝑓(𝑋))(𝑓(𝑥)), which implies 𝑓∘𝑖_𝑓(𝑋)=𝑖𝑑_𝑓(𝑋). By Remark 36 and Proposition 37, (𝑖_𝑓(𝑋),𝑓) is a fuzzy 𝑍-continuous section-retraction pair between (𝑓(𝑋),𝑒_𝑓(𝑋)) and (𝑋,𝑒). Since (𝑋,𝑒) is a fuzzy 𝑍-continuous poset, then (𝑓(𝑋),𝑒_𝑓(𝑋)) is a fuzzy 𝑍-continuous poset which follows from Theorem 30.

Definition 39. A fuzzy closure system on 𝑋 is fuzzy 𝑍-complete if for each 𝐴∈𝑍_𝐿(𝑀), such that ⊔_𝑋𝐴 exists, we have ⊔_𝑋𝐴∈𝑀 (in other words, 𝑀 is closed in 𝑋 under the formation of sups of fuzzy 𝑍-subsets of 𝑀).

Theorem 40. In a fuzzy 𝑍-complete poset (𝑋,𝑒), the one-to-one correspondence established by Theorem 32 induces a one-to-one correspondence between fuzzy 𝑍-complete closure systems on 𝑋 and fuzzy 𝑍-continuous closure operators on 𝑋.

Proof. Suppose that a subset 𝑀 is a fuzzy 𝑍-complete closure system on 𝑋; by Theorem 32, it suffices to show that 𝑐∶𝑋→𝑋 is fuzzy 𝑍-continuous. Since 𝑐 is a fuzzy monotone map, then for any 𝐴∈𝑍_𝐿(𝑋), 1=𝑒_𝑋(⊔_𝑋𝑐^→_𝐿(𝐴),𝑐(⊔_𝑋𝐴)). It remains to show 1=𝑒_𝑋(𝑐(⊔_𝑋𝐴),⊔_𝑋𝑐^→_𝐿(𝐴)), to this end, for any 𝑦∈𝑋, 𝑒_𝑋⊔_𝑋𝑐^→_𝐿(𝐴)=,𝑦_𝑥∈𝑀𝑐^→_𝐿(𝐴)(𝑥)⟶𝑒_𝑋=𝑥,𝑦_{𝑥∈𝑀𝑥′∈𝑋}𝐴𝑥^′∗𝑒_𝑀𝑥,𝑐𝑥^′⟶𝑒_𝑋=𝑥,𝑦_{𝑥′∈𝑋}𝐴𝑥^′⟶_𝑥∈𝑀𝑒_𝑀𝑥,𝑐𝑥^′⟶𝑒_𝑋=𝑥,𝑦_{𝑥′∈𝑋}𝐴𝑥^′⟶𝑒_𝑋𝑥^′,𝑐𝑥^′∗𝑒_𝑋𝑐𝑥^′≤,𝑦_{𝑥′∈𝑋}𝐴𝑥^′⟶𝑒_𝑋𝑥^′,𝑦=𝑒_𝑋⊔_𝑋𝐴,𝑦. (32) Thus for any 𝑦∈𝑋, 𝑒_𝑋(⊔_𝑋𝑐^→_𝐿(𝐴),𝑦)≤𝑒_𝑋(⊔_𝑋𝐴,𝑦), especially, set 𝑦=⊔_𝑋𝑐^→_𝐿(𝐴) and note that ⊔_𝑋𝑐^→_𝐿(𝐴)=⊔_𝑀𝑐^→_𝐿(𝐴)=𝑐(⊔_𝑋𝑐^→_𝐿(𝐴)); then 1=𝑒_𝑋⊔_𝑋𝑐^→_𝐿(𝐴),⊔_𝑋𝑐^→_𝐿(𝐴)≤𝑒_𝑋⊔_𝑋𝐴,⊔_𝑋𝑐^→_𝐿(𝐴)≤𝑒_𝑋𝑐⊔_𝑋𝐴,⊔_𝑋𝑐^→_𝐿(𝐴). (33)
Assume that 𝑐∶𝑋→𝑋 is a fuzzy 𝑍-continuous closure operator, then 𝑐(𝑋) is a fuzzy 𝑍-complete closure system which follows from Proposition 35 and Theorem 38.

By Theorems 32(2), 38, and 40, we can deduce the following result.

Theorem 41. Any fuzzy 𝑍-complete closure system of a fuzzy 𝑍-continuous poset is fuzzy 𝑍-continuous.

5. Fuzzy 𝑍-Algebraic Posets

In this section, the notion of fuzzy 𝑍-algebraic posets is given, then we investigate some algebraic properties of such a structure. Moreover, an extension theorem of a fuzzy 𝑍-algebraic poset is obtained.

In universal algebra, algebraic lattice has become familiar objects as lattices of congruences and lattices of subalgebras of an algebra. Yao [14] gave the definition of fuzzy (directed) algebraic posets, and Stubbe [21] presented a systematic investigation of fuzzy (directed) algebraic posets. Here, we study such an algebraic structure from the viewpoint of the fuzzy subset systems.

Definition 42. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset and 𝑥∈𝑋. Define a map 𝑘_𝑥∶𝑋→𝐿 by 𝑘_𝑥=↓𝑥|_𝐾(𝑋), 𝑘_𝑥 restricted on 𝐾(𝑋), that is, 𝑘_𝑥(𝑦)=𝑒(𝑦,𝑥) if 𝑦∈𝐾(𝑋) and otherwise 0. The fuzzy 𝑍-complete poset (𝑋,𝑒) is said to be a fuzzy 𝑍-algebraic poset if and only if 𝑘_𝑥∈𝑍_𝐿(𝑋) and 𝑥=⊔𝑘_𝑥.

Next we give an example of fuzzy 𝑍-algebraic posets.

Example 43. In a fuzzy union-complete subset system 𝑍_𝐿, (𝑍𝐼_𝐿(𝑋),sub) is a fuzzy 𝑍-algebraic poset.

Proof. Let Φ∈𝑍𝐼_𝐿(𝑍𝐼_𝐿(𝑋)). It is easy to verify that ⊔Φ=⋁_{𝐼∈𝑍𝐼𝐿(𝑋)}Φ(𝐼)∗𝐼. Since 𝑍_𝐿 is fuzzy union-complete, then ⊔Φ∈𝑍_𝐿(𝑋). Moreover, ⋁_{𝐼∈𝑍𝐼𝐿(𝑋)}Φ(𝐼)∗𝐼 is a fuzzy lower set; hence (𝑍𝐼_𝐿(𝑋),sub) is a fuzzy 𝑍-complete poset.
Next we show ↓𝑦∈𝐾(𝑍𝐼_𝐿(𝑋)) for any 𝑦∈𝑋. To this end, it suffices to show ⇓𝐼(𝐼)=⋁_𝑥∈𝑋𝐼(𝑥)∗sub(𝐼,↓𝑥) for any 𝐼∈𝑍𝐼_𝐿(𝑋).
On the one hand, note that (⊔Φ)(𝑥)=Φ(↓𝑥); then 𝐼(𝑥)∗sub(𝐼,↓𝑥)∗sub(𝐼,⊔Φ)≤sub()∗(𝑥)=𝐼,↓𝑥⊔Φ)(sub()()(𝐼).𝐼,↓𝑥∗Φ↓𝑥≤Φ (34) Thus 𝐼(𝑥)∗sub(𝐼,↓𝑥)≤sub(𝐼,⊔Φ)→Φ(𝐼), which implies that ⋁_𝑥∈𝑋𝐼(𝑥)∗sub(𝐼,↓𝑥)≤⇓𝐼(𝐼).
On the other hand, since 𝑓=↓∶𝑋→𝑍𝐼_𝐿(𝑋) is a fuzzy monotone map, then for any 𝐼∈𝑍𝐼_𝐿(𝑋), we have 𝑓^→_𝐿(𝐼)=⋁_𝑥∈𝑋𝐼(𝑥)∗sub(−,𝑓(𝑥))=⋁_𝑥∈𝑋𝐼(𝑥)∗sub(−,↓𝑥)∈𝑍𝐼_𝐿(𝑍𝐼_𝐿(𝑋)) and ⊔𝑓^→_𝐿(𝐼)=𝐼. Then ⇓𝐼(𝐼)_{Φ∈𝑍𝐼𝐿𝑍𝐼𝐿(𝑋)}=(sub()(𝐼≤𝐼,⊔Φ⟶Φ))sub𝐼,⊔𝑓^→_𝐿(𝐼)⟶𝑓^→_𝐿(𝐼𝐼).=)(_𝑥∈𝑋𝐼(𝑥)∗sub(𝐼,↓𝑥). (35) Hence ⇓𝐼(𝐼)=⋁_𝑥∈𝑋𝐼(𝑥)∗sub(𝐼,↓𝑥). This indicates that ↓𝑦∈𝐾(𝑍𝐼_𝐿(𝑋)).
At last, we show for any 𝐼∈𝑍𝐼_𝐿(𝑋), 𝑘_𝐼 satisfies Definition 42. It is easy to check that 𝑘_𝐼∈𝑍𝐼_𝐿(𝑋), and it remains to show ⊔𝑘_𝐼=𝐼. Appealing to the previous proof, we obtain ⊔𝑘_𝐼=_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑘_𝐼𝐼^′∗𝐼^′≥_𝑥∈𝑋sub(↓𝑥,𝐼)∗↓𝑥=𝐼,⊔𝑘_𝐼=_{𝐼′∈𝑍𝐼𝐿(𝑋)}𝑘_𝐼𝐼^′∗𝐼^′=_{𝐼′∈𝑍𝐼𝐿(𝑋)}sub𝐼^′,𝐼∗𝐼^′≤𝐼. (36) These complete the proof.

Definition 44. In a fuzzy 𝑍-complete poset, a fuzzy subset system 𝑍_𝐿 is said to be consistent if for any 𝑥∈𝑋, there exists a fuzzy subset 𝐴∈𝑍_𝐿(𝑋) satisfying that 𝐴≤⇓_𝑍𝑥 and 𝑥=⊔𝐴, then ⇓_𝑍𝑥∈𝑍_𝐿(𝑋) and 𝑥=⊔⇓_𝑍𝑥.

Remark 45. Fuzzy 𝒟-complete posets, fuzzy 𝒫-complete posets, and fuzzy ℒ-complete posets are consistent.

Proof. Obviously, fuzzy 𝒫-complete posets and fuzzy ℒ-complete posets are consistent. Next we give the proof in terms of fuzzy 𝒟-complete posets.
Let (𝑋,𝑒) be a fuzzy dcpo. For any 𝑥∈𝑋, if there exists a fuzzy directed subset 𝐴 such that 𝐴≤⇓𝑥 and 𝑥=⊔𝐴, we should show that ⇓𝑥∈𝒟_𝐿(𝑋) and 𝑥=⊔⇓𝑥. For any 𝑦∈𝑋, we first show ⇓𝑥(𝑦)≤⋁_𝑑∈𝑋𝐴(𝑑)∗𝑒(𝑦,𝑑). Indeed, ⇓𝑥𝑦=_{𝐼∈ℐ𝐿(𝑋)}𝑒()𝑦()⟶𝑥,⊔𝐼⟶𝐼≤𝑒𝑥,⊔↓𝐴_𝑑∈𝑋𝐴(𝑑)∗𝑒𝑦,𝑑=1⟶_𝑑∈𝑋𝐴(𝑑)=∗𝑒𝑦,𝑑_𝑑∈𝑋𝐴(𝑑)∗𝑒𝑦,𝑑. (37) Thus, for any 𝑎,𝑏∈𝑋, we have ⇓𝑥(𝑎)∗⇓𝑥(𝑏)≤_{𝑑1,𝑑2∈𝑋}𝐴𝑑₁∗𝐴𝑑₂∗𝑒𝑎,𝑑₁∗𝑒𝑏,𝑑₂≤_{𝑑1,𝑑2∈𝑋𝑑∈𝑋}𝐴(𝑑)∗𝑒𝑑₁,𝑑∗𝑒𝑑₂,𝑑∗𝑒𝑎,𝑑₁∗𝑒𝑏,𝑑₂≤_{𝑑1,𝑑2∈𝑋𝑑∈𝑋}𝐴(𝑑)()()=∗𝑒𝑎,𝑑∗𝑒𝑏,𝑑_𝑑∈𝑋𝐴(𝑑)()()≤∗𝑒𝑎,𝑑∗𝑒𝑏,𝑑_𝑑∈𝑋⇓𝑥(𝑑)∗𝑒(𝑎,𝑑)∗𝑒(𝑏,𝑑). (38) Moreover, ⋁_𝑦∈𝑋⇓𝑥(𝑦)=1 follows from 1=⋁_𝑦∈𝑋𝐴(𝑦)≤⋁_𝑦∈𝑋⇓𝑥(𝑦). Hence, ⇓𝑥 is fuzzy directed.
Since 𝐴≤⇓𝑥 and ⊔ is fuzzy monotone, then 1=sub(𝐴,⇓𝑥)≤𝑒(⊔𝐴,⊔⇓𝑥)=𝑒(𝑥,⊔⇓𝑥). Meanwhile, 1=sub(⇓𝑥,↓𝑥)≤𝑒(⊔⇓𝑥,⊔↓𝑥)=𝑒(⊔⇓𝑥,𝑥). Therefore, 𝑥=⊔⇓𝑥.

Lemma 46. In a fuzzy union-complete subset system 𝑍_𝐿, if (𝑋,𝑒) is a fuzzy 𝑍-algebraic poset, then for any 𝑥∈𝑋, we have ⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧∈𝑍_𝐿(𝑋).

Proof. By Definition 13(3), ↓𝑥∈𝑍_𝐿(𝑋) for any 𝑥∈𝑋. Therefore, the map 𝑓=↓∶𝑋→𝑍_𝐿(𝑋) is well defined. Assume that (𝑋,𝑒) is a fuzzy 𝑍-algebraic poset, then 𝑘_𝑥∈𝑍_𝐿(𝑋). Since the map 𝑓=↓ is fuzzy monotone, by Definition 13(2), 𝑓^→_𝐿(𝑘_𝑥)=⋁_{𝑧∈𝐾(𝑋)}𝑘_𝑥(𝑧)∗sub(−,↓𝑧)∈𝑍_𝐿(𝑍_𝐿(𝑋)). Since 𝑍_𝐿 is fuzzy union-complete, to show ⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧∈𝑍_𝐿(𝑋), it suffices to show that ⊔(⋁_{𝑧∈𝐾(𝑋)}𝑘_𝑥(𝑧)∗sub(−,↓𝑧))=⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧. For any 𝜓∈𝑍_𝐿(𝑋), sub_{(𝑋)𝑧∈𝐾}(𝑧)=↓𝑥∗↓𝑧,𝜓_{𝑦∈𝑋(𝑋)𝑧∈𝐾}(𝑧)𝑦𝑦=↓𝑥∗↓𝑧⟶𝜓_{(𝑋)𝑧∈𝐾}↓𝑥(𝑧)⟶sub=↓𝑧,𝜓_{𝜑∈𝑍𝐿(𝑋)(𝑋)𝑧∈𝐾}↓𝑥(𝑧)∗sub𝜑,↓𝑧⟶sub=𝜑,𝜓_{𝜑∈𝑍𝐿(𝑋)}𝑓^→_𝐿𝑘_𝑥𝜑⟶sub𝜑,𝜓. (39) Thus ⊔𝑓^→_𝐿(𝑘_𝑥)=⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧.

Proposition 47. In a fuzzy union-complete and consistent subset system 𝑍_𝐿, if (𝑋,𝑒) is a fuzzy 𝑍-algebraic poset, then (𝑋,𝑒) is a fuzzy 𝑍-continuous poset and for any 𝑥,𝑦∈𝑋, ⇓_𝑍𝑥𝑦=_{(𝑋)𝑧∈𝐾}↓𝑥(𝑧)∗↓𝑧𝑦. (40)

Proof. The 𝑍-continuity of (𝑋,𝑒) is immediate from Definition 44; it remains to show that for any 𝑥,𝑦∈𝑋, ⇓_𝑍𝑥(𝑦)=⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧(𝑦).
On the one hand, by Proposition 17(2), ↓𝑥(_{(𝑋)𝑧∈𝐾}𝑧)∗↓𝑧𝑦=_{(𝑋)𝑧∈𝐾}⇓_𝑍𝑧(𝑧)∗↓𝑥(𝑧)∗↓𝑧𝑦≤⇓_𝑍𝑥𝑦. (41)
On the other hand, since 𝑍_𝐿 is fuzzy union-complete and ⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧 is a fuzzy lower set, by Lemma 46, ⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧∈𝑍𝐼_𝐿(𝑋). It is easy to check that 𝑥=⊔(⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧), then ⇓_𝑍𝑥𝑦=_{𝐼∈𝑍𝐼𝐿(𝑋)}𝑒(𝑥,⊔𝐼)⟶𝐼𝑦≤𝑒𝑥,⊔_{(𝑋)𝑧∈𝐾}↓𝑥(𝑧)∗↓𝑧⟶_{(𝑋)𝑧∈𝐾}(𝑧)𝑦=↓𝑥∗↓𝑧_{(𝑋)𝑧∈𝐾}↓𝑥(𝑧)∗↓𝑧𝑦. (42) Therefore, ⇓_𝑍𝑥(𝑦)=⋁_{𝑧∈𝐾(𝑋)}↓𝑥(𝑧)∗↓𝑧(𝑦).

Proposition 48. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset and 𝑐∶(𝑋,𝑒)→(𝑋,𝑒) a fuzzy 𝑍-continuous closure operator. Then 𝑐(𝐾(𝑋))⊆𝐾(𝑐(𝑋)) and the equality holds if (𝑋,𝑒) is a fuzzy 𝑍-algebraic poset.

Proof. For any 𝑥∈𝐾(𝑋), 𝑒(𝑥,𝑐(𝑥))=1 and for any 𝐼^′∈𝑍𝐼_𝐿(𝑐(𝑋)), by Proposition 35, we have ⊔𝐼^′∈𝑐(𝑋). Note that 𝑖^→_{𝑐(𝑋)𝐿}(𝐼^′)∈𝑍𝐼_𝐿(𝑋); then ⇓_𝑍𝑥(𝑥)=_{𝐼∈𝑍𝐼𝐿(𝑋)}(𝑒()(𝑥≤𝑥,⊔𝐼⟶𝐼))_{𝐼′∈𝑍𝐼𝐿(𝑐(𝑋))}𝑒_𝑋𝑥,⊔𝑖^→_{𝑐(𝑋)𝐿}𝐼^′⟶𝑖^→_{𝑐(𝑋)𝐿}𝐼^′𝑥)=_{𝐼′∈𝑍𝐼𝐿(𝑐(𝑋))}(𝑒_𝑋𝑥,⊔𝐼^′⟶_{𝑑′(𝑋)∈𝑐}𝐼^′𝑑^′∗𝑒_𝑋𝑥,𝑑^′≤_{𝐼′∈𝑍𝐼𝐿(𝑐(𝑋))}𝑒_𝑐(𝑋)𝑐(𝑥),⊔𝐼^′∗𝑒_𝑋((𝑥⟶𝑥,𝑐))_{𝑑′(𝑋)∈𝑐}𝐼^′𝑑^′∗𝑒_𝑐(𝑋)𝑐(𝑥),𝑑^′=_{𝐼′∈𝑍𝐼𝐿(𝑐(𝑋))}𝑒_𝑐(𝑋)𝑐(𝑥),⊔𝐼^′⟶𝐼^′(𝑐(𝑥))=⇓_𝑍𝑐(𝑥)(𝑐(𝑥)). (43)
Further, if (𝑋,𝑒) is a fuzzy 𝑍-algebraic poset, then for any 𝑦∈𝐾(𝑐(𝑋)), there exists 𝑘_𝑦∈𝑍_𝐿(𝑋) such that 𝑦=⊔𝑘_𝑦. Since 𝑐 is fuzzy monotone, then 𝑐^→_𝐿(𝑘_𝑦)∈𝑍𝐼_𝐿(𝑐(𝑋)). Consider 1=⇓_𝑐(𝑋)𝑦𝑦=_{𝐼∈𝑍𝐼𝐿(𝑐(𝑋))}𝑒_𝑐(𝑋)𝑦,⊔𝐼⟶𝐼𝑦≤𝑒_𝑐(𝑋)𝑦,⊔𝑐^→_𝐿𝑘_𝑦⟶𝑐^→_𝐿𝑘_𝑦𝑦=𝑐^→_𝐿𝑘_𝑦𝑦=_{(𝑋)𝑥∈𝐾}𝑒(𝑥≤𝑥,𝑦∗𝑒𝑦,𝑐)_{(𝑋)𝑥∈𝐾}𝑒𝑐(𝑥),𝑦∗𝑒𝑦,𝑐(𝑥), (44) which indicates that there exists 𝑥^′∈𝐾(𝑋) such that 1=𝑒(𝑐(𝑥^′),𝑦)∗𝑒(𝑦,𝑐(𝑥^′)), hence 𝑦=𝑐(𝑥^′). Therefore, 𝑦∈𝑐(𝐾(𝑋)).

Proposition 49. Let (𝑋,𝑒) be a fuzzy 𝑍-algebraic poset and 𝑐∶(𝑋,𝑒)→(𝑋,𝑒) a fuzzy 𝑍-continuous closure operator. Then (𝑐(𝑋),𝑒_𝑐(𝑋)) is a fuzzy 𝑍-algebraic poset (relative to the induced fuzzy order).

Proof. For any 𝑦∈𝑐(𝑋)⊆𝑋, there exists 𝑘_𝑦∈𝑍_𝐿(𝑋) such that 𝑦=⊔𝑘_𝑦. Note that 𝑐∶(𝑋,𝑒)→(𝑋,𝑒) is a fuzzy 𝑍-continuous closure operator; then 𝑦=𝑐𝑦=𝑐⊔𝑘_𝑦=⊔𝑐^→_𝐿𝑘_𝑦. (45)
To show (𝑐(𝑋),𝑒_𝑐(𝑋)) is a fuzzy 𝑍-algebraic poset, it suffices to show that 𝑐^→_𝐿(𝑘_𝑦)=𝑘_𝑦. On the one hand, for any 𝑧∈𝐾(𝑐(𝑋)), 𝑐^→_𝐿𝑘_𝑦(𝑧)=_{(𝑋)𝑥∈𝐾}𝑘_𝑦(𝑥)((𝑥≤∗𝑒𝑧,𝑐))_{(𝑋)𝑥∈𝐾}𝑒𝑥,⊔𝑘_𝑦((𝑥≤∗𝑒𝑧,𝑐))_{(𝑋)𝑥∈𝐾}𝑒𝑐(𝑥),𝑦∗𝑒(𝑧,𝑐(𝑥))≤𝑒𝑧,𝑦=𝑘_𝑦(𝑧). (46)
On the other hand, for 𝑧∈𝐾(𝑐(𝑋)), by Proposition 48, there exists 𝑧^′∈𝐾(𝑋) with 𝑐(𝑧^′)=𝑧. Since 𝑐 is a fuzzy closure operator, by Proposition 12, (𝑖_𝑐(𝑋),𝑐) is a fuzzy Galois connection between 𝑐(𝑋) and 𝑋. Thus 𝑘_𝑦(𝑧)=𝑒(𝑧,𝑦)=𝑒(𝑐(𝑧^′),𝑦)=𝑒(𝑧^′,𝑖_𝑐(𝑋)(𝑦))=𝑒(𝑧^′,𝑦), and 𝑐^→_𝐿𝑘_𝑦(𝑧)=_{(𝑋)𝑥∈𝐾}𝑒((𝑥𝑧𝑥,𝑦∗𝑒𝑧,𝑐))≥𝑒^′,𝑦∗𝑒𝑧,𝑐𝑧^′=𝑒𝑧^′,𝑦. (47)

By Theorem 32 and Proposition 49, we get the following.

Theorem 50. Any 𝑍-complete closure system of a fuzzy 𝑍-algebraic poset is a fuzzy 𝑍-algebraic poset.

Lemma 51. Let (𝑋,𝑒) be a fuzzy 𝑍-complete poset and 𝐴∈𝑍_𝐿(𝑋). Then 𝐾_𝐴=𝑘_⊔𝐴, where 𝐾_𝐴∶𝑋→𝐿 defined by 𝐾_𝐴=⋁_{𝑥′∈𝑋}𝐴(𝑥^′)∗↓𝑥^′|_𝐾(𝑋) for any 𝐴∈𝑍_𝐿(𝑋), 𝐾_𝐴 restricted on 𝐾(𝑋), that is, 𝐾_𝐴(𝑦)=⋁_{𝑥′∈𝑋}𝐴(𝑥^′)∗𝑒(𝑦,𝑥^′) if 𝑦∈𝐾(𝑋) and otherwise 0.

Proof. For any 𝑥∈𝐾(𝑋), 𝐾_𝐴(𝑥)=_𝑦∈𝑋𝐴𝑦∗𝑒𝑥,𝑦≤_𝑦∈𝑋𝑒()𝑦,⊔𝐴∗𝑒𝑥,𝑦=𝑒𝑥,⊔𝐴=𝑘_⊔𝐴(𝑥). (48)
For the converse, 𝑘_⊔𝐴(𝑥)=⇓_𝑍(⊔𝐴)(𝑥)=_{𝐼∈𝑍𝐼𝐿(𝑋)}(𝑒()(𝑥≤⊔𝐴,⊔𝐼⟶𝐼))_𝑑∈𝑋𝐴(𝑑)∗𝑒(𝑥,𝑑)=𝐾_𝐴(𝑥). (49)

Lai and Zhang [22] pointed out that for any dcpo (𝑌,𝑒) and fuzzy monotone map 𝑓∶(𝑋,𝑒)→(𝑌,𝑒), there is a unique fuzzy Scott continuous map 𝑓∶(ℐ(𝑋),𝑒)→(𝑌,𝑒) such that 𝑓=𝑓∘𝐲, where 𝐲 is a yoneda embedding. The following is the similar conclusion of that. It says that any fuzzy monotone maps defined on the compact elements of fuzzy 𝑍-algebraic posets extend uniquely to fuzzy 𝑍-continuous maps on the whole fuzzy poset.

Theorem 52. Let (𝑋,𝑒) be a fuzzy 𝑍-algebraic poset, (𝑌,𝑒) a fuzzy 𝑍-complete poset, and 𝑓∶𝐾(𝑋)→𝑌 a fuzzy monotone map. Then there exists a unique fuzzy 𝑍-continuous 𝑓∶𝑋→𝑌, which extends 𝑓.

Proof. For any 𝑥∈𝑋, there exists 𝑘_𝑥∈𝑍_𝐿(𝑋) such that 𝑥=⊔𝑘_𝑥. Since 𝑓 is fuzzy monotone and (𝑌,𝑒) a fuzzy 𝑍-complete poset, then ⊔𝑓^→_𝐿(𝑘_𝑥) exists in 𝑌. Now for 𝑓 to extend 𝑓 to be fuzzy 𝑍-continuous, we must have 𝑓(⊔𝐴)=⊔𝑓^→_𝐿(𝐴) for any 𝐴∈𝑍_𝐿(𝑋). We need to show that 𝑓 defined in this way is indeed well defined, fuzzy 𝑍-continuous.
For any 𝑥∈𝐾(𝑋), 𝑓(𝑥)=𝑓(⊔𝑘_𝑥)=⊔𝑓^→_𝐿(𝑘_𝑥). Next we show that ⊔𝑓^→_𝐿(𝑘_𝑥)=𝑓(𝑥). For any 𝑦∈𝑌, 𝑓^→_𝐿𝑘_𝑥𝑦=_{(𝑋)𝑧∈𝐾}𝑘_𝑥(𝑧)(𝑧)≤∗𝑒𝑦,𝑓_{(𝑋)𝑧∈𝐾}𝑒𝑓(𝑧)(𝑥)(𝑧)(𝑥),,𝑓∗𝑒𝑦,𝑓≤𝑒𝑦,𝑓 (50) and for any 𝑧∈𝑌, _𝑦∈𝑌𝑓^→_𝐿𝑘_𝑥𝑦=⟶𝑒𝑦,𝑧_{𝑦∈𝑌𝑧′(𝑋)∈𝐾}𝑘_𝑥𝑧^′∗𝑒𝑦,𝑓𝑧^′=⟶𝑒𝑦,𝑧_{𝑧′(𝑋)∈𝐾}𝑘_𝑥𝑧^′⟶𝑒𝑓𝑧^′,𝑧≤𝑘_𝑥(𝑥)𝑓(𝑥)𝑓(𝑥).⟶𝑒,𝑧=𝑒,𝑧 (51) Hence by Definition 5, 𝑓(𝑥)=⊔𝑓^→_𝐿(𝑘_𝑥); that is, 𝑓(𝑥)=𝑓(𝑥) holds for all 𝑥∈𝐾(𝑋). Moreover, for 𝑥∈𝑋, 𝑓(𝑥)=⊔𝑓^→_𝐿(𝑘_𝑥), and these indicate that 𝑓 is well defined.
Next we show that 𝑓 is fuzzy 𝑍-continuous. Since 𝑓 is a fuzzy monotone map, then for any 𝑥,𝑦∈𝑋, 𝑒𝑥,𝑦≤sub𝑘_𝑥,𝑘_𝑦≤sub𝑓^→_𝐿𝑘_𝑥,𝑓^→_𝐿𝑘_𝑦≤sub⊔𝑓^→_𝐿𝑘_𝑥,⊔𝑓^→_𝐿𝑘_𝑦=𝑒𝑓(𝑥),𝑓𝑦, (52) which implies that 𝑓 is fuzzy monotone; hence we have 1=𝑒(⊔𝑓^→_𝐿(𝐴),𝑓(⊔𝐴)), and it remains to show the other side. To this end, by Lemma 51, 𝑓(⊔𝐴)=⊔𝑓^→_𝐿(𝑘_⊔𝐴)=⊔𝑓^→_𝐿(𝐾_𝐴), and note that 𝑓(𝑥)=𝑓(𝑥) for all 𝑥∈𝐾(𝑋), it suffices to show that 𝑓^→_𝐿(𝐾_𝐴)≤𝑓^→_𝐿(𝐴). For any 𝑦∈𝑌, 𝑓^→_𝐿𝐾_𝐴𝑦=_{(𝑋)𝑥∈𝐾}𝐾_𝐴(𝑥)(𝑥)=∗𝑒𝑦,𝑓_{(𝑋)𝑥∈𝐾𝑥′∈𝑋}𝐴𝑥^′∗𝑒𝑥,𝑥^′(𝑥)≤∗𝑒𝑦,𝑓_{(𝑋)𝑥∈𝐾𝑥′∈𝑋}𝐴𝑥^′∗𝑒𝑓(𝑥),𝑓𝑥^′∗𝑒𝑦,𝑓(𝑥)≤_{𝑥′∈𝑋}𝐴𝑥^′∗𝑒𝑦,𝑓𝑥^′=𝑓^→_𝐿(𝐴)𝑦. (53) These indicate that 𝑓 is fuzzy 𝑍-continuous and the uniqueness of 𝑓 is obvious.
Thus, we complete the proof.

6. Conclusions

Through introducing the concept of fuzzy subset systems, we study fuzzy 𝑍-continuous posets, strongly fuzzy 𝑍-continuous posets, and fuzzy 𝑍-algebraic posets. Because of the outstanding properties of complete residuated lattices, the fuzzy 𝑍-continuous posets and fuzzy 𝑍-algebraic posets inherit many good properties from 𝑍-continuous posets and 𝑍-algebraic posets, such as the fuzzy 𝑍-continuous section-retraction pair between fuzzy 𝑍-continuous posets, and we also give the definition of fuzzy 𝑍-complete closure systems and associate a fuzzy 𝑍-continuous closure operator with a fuzzy 𝑍-complete closure system; it is shown that each fuzzy 𝑍-complete closure system of a fuzzy 𝑍-continuous poset is fuzzy 𝑍-continuous; we also present an extension theorem of fuzzy 𝑍-algebraic posets.

Acknowledgment

This work was supported by the National Natural Science Foundation of China, Grant no. 11071061.