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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 607934, 14 pages

http://dx.doi.org/10.1155/2013/607934

## Fuzzy -Continuous Posets

^{1}College of Mathematics and Econometrics, Hunan University, Changsha 410082, China^{2}Faculty of Sciences, Nanchang Institute of Technology, Nanchang 330099, China

Received 27 April 2013; Revised 7 July 2013; Accepted 8 July 2013

Academic Editor: Stefan Siegmund

Copyright © 2013 S. P. Rao and Q. G. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 -continuous posets and fuzzy -algebraic posets are given. Furthermore, we try to find some natural classes of fuzzy -continuous maps under which the images of such fuzzy algebraic structures can be preserved; we also think about fuzzy -continuous closure operators in alternative ways. An extension theorem is presented for extending a fuzzy monotone map defined on the -compact elements to a fuzzy -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 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 -inductive posets, which is really a generalization of 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 -subsets, where 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 -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 -categories.

In view of the increasing interest in quantitative domain theory and -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 -continuous posets and fuzzy strongly -continuous posets, and study the relationship between such algebraic structures. We also discuss the fuzzy -continuous section-retraction pair between fuzzy -continuous posets. In Section 4, we introduce the concept of fuzzy -complete closure systems and associate a fuzzy -continuous closure operator with a fuzzy -complete closure system. We prove that each fuzzy -complete closure system of a fuzzy -continuous poset is fuzzy -continuous. In Section 5, the notion of fuzzy -algebraic posets is given, then some algebraic properties of such a structure are studied. An extension theorem based on -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, always denotes a complete residuated lattice.

*Definition 1. *A complete residuated lattice is an algebraic structure such that(1) is a complete lattice with the least element and the greatest element ;(2) is a commutative monoid; that is, is commutative, associative, and holds for all ;(3) and form an adjoint pair, that is, for any , .

Proposition 2. *For a complete residuated lattic , one has*(1)* and **,*(2)*,*(3)*,*(4)*, and hence ** whenever **,*(5)*, and hence ** whenever **,*(6)*,*(7)*,*(8)*,*(9)*.*

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

Let be a nonempty set. An -subset on is a map from to , and the family of all -subsets on will be denoted by . All algebraic operations on can be extended pointwisely to the power set . That is, for any , we have , and .

*Definition 3 (see [13, 25]). *A fuzzy poset is a pair such that is a non-empty set and is a map, called a fuzzy order, that satisfies for any ,(1);(2);(3).

If is an -partial order on , then is called an -partial ordered set (simply, a fuzzy poset). To study fuzzy relational systems, Belohlávek [23] defined and studied an -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 , the subsethood degree map is defined by for each pair , , then is a fuzzy partial order on and is a fuzzy poset. Especially, when , we have .

(2) If , then is also a fuzzy poset (relative to the induced order from ), where is the restriction of to .

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 , an element is called a join (or meet) of a fuzzy subset , in symbols or if(1)for any , or ,(2)for any , or .

For any , or is defined as for any , or . For , is defined as . is called a fuzzy upper set (or a fuzzy lower set) if for any , or .

Proposition 6. *Let be a fuzzy poset. Then*(1)* if and only if for any **, **,*(2)* if and only if for any **, **,** particularly, when , one has
*

is called the maximal (or minimal) element of , in symbols (or ) if and for all , (or ). It is easy to check that if has a maximal (or minimal) element, then it is unique.

*Definition 7. *Given two fuzzy posets and , a map is said to be fuzzy monotone if for any , . Furthermore, a fuzzy monotone map is called a projection on if for all , . We say a projection is a fuzzy closure operator if for all .

*Definition 8. *Let and be two fuzzy posets, and two fuzzy monotone maps. The pair is called a fuzzy Galois connection between and provided that
where is called the upper adjoint of and dually is the lower adjoint of .

Obviously, a fuzzy Galois connection is an extension of a crisp Galois connection. The crisp Galois connection is defined as follows: for any , , and its relative properties can be found in [30].

*Example 9. *Let be a map. For each , let . Then we obtain a powerset operator: . Conversely, define a powerset operator by
Then is a fuzzy Galois connection between and .

Proposition 10. *Let and be two fuzzy posets, and two maps. Then the following are equivalent:*(1)* is a fuzzy Galois connection,*(2)* is fuzzy monotone and ** for all **,*(3)* is fuzzy monotone and ** for all **.*

Proposition 11. *Let and be two fuzzy posets, and two maps.*(1)*If ** is a fuzzy monotone map and has a lower adjoint, then for any ** such that ** exists, **.*(2)*If ** is a fuzzy monotone map and has an upper adjoint, then for any ** such that ** exists, **.*

For any map , we denote the corestriction to the image as and the inclusion of the image into accordingly as . Thus, each has the decomposition . If , then is the restriction and corestriction .

Proposition 12. *Let be a fuzzy poset and a fuzzy monotone map. Then the following are equivalent:*(1)* is a fuzzy closure operator,*(2)* is a fuzzy Galois connection between ** and **,*(3)*There is a fuzzy Galois connection ** between some fuzzy posets ** and ** such that **.*

#### 3. Fuzzy Subset Systems and Fuzzy -Continuous Posets

In this section, we present the definition of fuzzy subset systems, and in such a framework we propose the notions of fuzzy -continuous posets and fuzzy strongly -continuous posets and study the relationship between them. We finally discuss the fuzzy -continuous section-retraction pair between fuzzy -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 satisfying the following conditions.(1)For any fuzzy poset , .(2)If and are two fuzzy posets and a fuzzy monotone map, then for any , .(3)For any , .

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

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

Based on a commutative, unital quantale , 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 is a fuzzy lower set and , but is not required.

*Definition 14. *A fuzzy poset is said to be fuzzy -complete if for any , exists. A fuzzy subset of a fuzzy poset is called a fuzzy -ideal of if it is a fuzzy lower set generated by some ; that is, for any fuzzy -ideal , there exists with . The sets of all fuzzy -subsets and all fuzzy -ideals on are denoted by and , respectively.

*Definition 15. *If is a fuzzy -complete poset, then for any , define by

is called the fuzzy -way-below relation. For , if , then we call a compact element in , and all compact elements in are denoted by .

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

Proposition 16. *Let be a fuzzy -complete poset. Then for any ,
**
That is, .*

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

Proposition 17. *Let be a fuzzy -complete poset. For any , then*(1)*; *(2)*.*

*Proof. *(1) By Definition 13(3), for any , , then

(2) Is straightforward.

*Definition 18. *A fuzzy -complete poset is called a fuzzy -continuous poset if for any , or and . A fuzzy -continuous poset is called strongly fuzzy -continuous if the fuzzy -way-below relation has the interpolation property: for any , .

Fuzzy -continuous posets are known in the literature as fuzzy domains. See [11–15, 18, 19, 25, 29] for further details. Fuzzy -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 is said to be fuzzy union-complete if for any , exists and .

*Remark 20. *The fuzzy subset systems , , and are fuzzy union-complete.

*Proof. *It is trivial that the statement holds for and . We now give the proof in terms of . Recall that a fuzzy subset is a fuzzy directed subset if , and for any , .

For any , put . We now show that and . For each ,
and for any ,

Lemma 21. *In a fuzzy union-complete subset system , if is a fuzzy -continuous poset, then for any , we have .*

*Proof. *Since is fuzzy -continuous, then for all . Hence the map is welldefined. It is easy to verify that is fuzzy monotone, then by Definition 13(2), for any , we have . Since is fuzzy union-complete, to show , it suffices to show that . To this end, for any ,
which indicates that .

Lemma 22. *In a fuzzy union-complete subset system , if is a fuzzy -continuous poset, and for any , set , then holds.*

*Proof. *We first show that . For any ,

On the one hand, for any , by Proposition 17(2),

On the other hand, since the fuzzy subset system is fuzzy union-complete, by Lemma 21, we have
Therefore, .

Proposition 23. *Let a fuzzy subset system be fuzzy union-complete. If is a fuzzy -continuous poset, then for any ,
*

*Proof. *For any , , then by Lemma 22,
This implies that .

By the arbitrariness of , we have

Conversely, by Proposition 17(1),
Thus
Hence, .

Theorem 24. *In a fuzzy union-complete subset system , a fuzzy -continuous poset is exactly a strongly fuzzy -continuous poset.*

*Proof. *By Definition 18, it suffices to show for all . 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,

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 -cocomplete -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 -continuous posets by using the given ones as building blocks.

Theorem 25. *Let be a fuzzy -complete poset. Then the following statements are equivalent:*(1)* is a fuzzy -continuous poset;*(2)*for each , is the smallest fuzzy -ideal with ;*(3)*for each , there exists a smallest fuzzy -ideal with ;*(4)*the sup map has a lower adjoint. These conditions imply*(5)*the sup map preserves all existing inf.*

*Proof. *: Condition holds if and only if for each , and . Hence is a fuzzy -ideal with .

Moreover, for all , with , we have
This establishes .

Condition trivially implies .

: It is easy to check that is a fuzzy monotone map. By Proposition 10, has a lower adjoint if and only if exists for all , where . This means is precisely the smallest fuzzy -ideal with .

: Let be the lower adjoint of . To show that is a fuzzy -continuous poset, it suffices to prove for all .

On the one hand, for all , since , then

On the other hand, ,

: 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 -continuous posets.

*Definition 26 (see [29]). *Let and be two fuzzy posets, and two fuzzy monotone maps. The pair is called a fuzzy monotone section-retraction pair if . In this situation, we call a fuzzy monotone retraction of .

It is clear that is surjective and is injective in a fuzzy monotone section-retraction pair .

*Definition 27. *Given two fuzzy -complete posets and , a fuzzy monotone map is said to be fuzzy -continuous if for any , .

*Definition 28. *A fuzzy monotone section-retraction pair is called a fuzzy -continuous section-retraction pair if and are fuzzy -complete posets and are both fuzzy -continuous, then we speak of as a fuzzy -continuous retraction of .

Lemma 29. *Let the pair be a fuzzy -continuous section-retraction-pair between and . Then for any , , .*

*Proof. *Note that for any , , then

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

*Proof. *Assume that is a fuzzy -continuous section-retraction-pair between and , where is a fuzzy -continuous poset, we need to show that is also a fuzzy -continuous poset.

Since , then for any , . Note that is fuzzy -continuous and is a fuzzy -continuous poset, then and
Therefore, to prove is fuzzy -continuous, it suffices to show that . On the one hand, for any and by Lemma 29,
On the other hand,
Therefore, .

#### 4. Fuzzy -Complete Closure Systems

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

*Definition 31 (see [33]). *If is a fuzzy poset, a fuzzy closure system on is a subset of such that for each , exists, where is the constant fuzzy set with the value on .

Theorem 32 (see [33]). *Let be a fuzzy poset, a fuzzy monotone map, and the corestriction to the image. Then one has the following.*(1)*If is a fuzzy closure system on and for each , , then the map is a fuzzy closure operator.*(2)*If is a fuzzy closure operator, then is a fuzzy closure system on .*(3)*The map defined by from the set of fuzzy closure systems on to the set of fuzzy closure operators on is bijective, and its converse is the map defined by .*

Lemma 33. *Let be a fuzzy -complete poset and . Then for any , . Here, is an inclusion map from to . Moreover, if , then .*

*Proof. *It is trivial that is fuzzy monotone, then for any , and exists. For any ,
Thus .

Furthermore, if , then for any , we have
This shows that . Hence .

Lemma 34. *Let be a fuzzy poset, a projection, and an inclusion map. Then for any , ; for any , .*

*Proof. * For any , note that ,

For the converse,

Similarly, we can prove .

Proposition 35. *Let be a fuzzy -complete poset and a fuzzy -continuous projection. Then is a fuzzy -complete poset.*

*Proof. *By Lemma 33, it suffices to show that for each . Actually, assume that , which means for all , . Then
which implies that .

Since is fuzzy -continuous and by Lemmas 33, 34, we have

*Remark 36. *The preceding proposition states that for any fuzzy -continuous projection , the inclusion map from to is fuzzy -continuous.

*Proof. *Since for any , and , then by Lemma 33 and Proposition 35, we have ; that is, the inclusion map from to is fuzzy -continuous.

Proposition 37. *Let be a fuzzy -complete poset, a fuzzy -continuous projection and . Then for all , (in other words, the corestriction of to is fuzzy -continuous).*

*Proof. *Since is fuzzy -continuous, then for any , . Moreover, is fuzzy monotone, which indicates that . By Lemma 33 and Proposition 35, . Thus .

Theorem 38. *Let be a fuzzy -continuous poset and a fuzzy -continuous projection. Then is fuzzy -continuous (relative to the induced fuzzy order).*

*Proof. *By Proposition 35, is a fuzzy -complete lattice. For any , , which implies . By Remark 36 and Proposition 37, is a fuzzy -continuous section-retraction pair between and . Since is a fuzzy -continuous poset, then is a fuzzy -continuous poset which follows from Theorem 30.

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

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

*Proof. *Suppose that a subset is a fuzzy -complete closure system on ; by Theorem 32, it suffices to show that is fuzzy -continuous. Since is a fuzzy monotone map, then for any , . It remains to show , to this end, for any ,
Thus for any , , especially, set and note that ; then

Assume that is a fuzzy -continuous closure operator, then is a fuzzy -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 -complete closure system of a fuzzy -continuous poset is fuzzy -continuous.*

#### 5. Fuzzy -Algebraic Posets

In this section, the notion of fuzzy -algebraic posets is given, then we investigate some algebraic properties of such a structure. Moreover, an extension theorem of a fuzzy -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 be a fuzzy -complete poset and . Define a map by , restricted on , that is, if and otherwise . The fuzzy -complete poset is said to be a fuzzy -algebraic poset if and only if and .

Next we give an example of fuzzy -algebraic posets.

*Example 43. *In a fuzzy union-complete subset system , is a fuzzy -algebraic poset.

*Proof. *Let . It is easy to verify that . Since is fuzzy union-complete, then . Moreover, is a fuzzy lower set; hence is a fuzzy -complete poset.

Next we show for any . To this end, it suffices to show for any .

On the one hand, note that ; then
Thus , which implies that .

On the other hand, since is a fuzzy monotone map, then for any , we have and . Then
Hence . This indicates that .

At last, we show for any , satisfies Definition 42. It is easy to check that , and it remains to show . Appealing to the previous proof, we obtain
These complete the proof.

*Definition 44. *In a fuzzy -complete poset, a fuzzy subset system is said to be consistent if for any , there exists a fuzzy subset satisfying that and , then and .

*Remark 45. *Fuzzy -complete posets, fuzzy -complete posets, and fuzzy -complete posets are consistent.

*Proof. *Obviously, fuzzy -complete posets and fuzzy -complete posets are consistent. Next we give the proof in terms of fuzzy -complete posets.

Let be a fuzzy dcpo. For any , if there exists a fuzzy directed subset such that and , we should show that and . For any , we first show . Indeed,
Thus, for any , we have
Moreover, follows from . Hence, is fuzzy directed.

Since and is fuzzy monotone, then . Meanwhile, . Therefore, .

Lemma 46. *In a fuzzy union-complete subset system , if is a fuzzy -algebraic poset, then for any , we have .*

*Proof. *By Definition 13(3), for any . Therefore, the map is well defined. Assume that is a fuzzy -algebraic poset, then . Since the map is fuzzy monotone, by Definition 13(2), . Since is fuzzy union-complete, to show , it suffices to show that . For any ,
Thus .

Proposition 47. *In a fuzzy union-complete and consistent subset system , if is a fuzzy -algebraic poset, then is a fuzzy -continuous poset and for any ,
*

*Proof. *The -continuity of is immediate from Definition 44; it remains to show that for any , .

On the one hand, by Proposition 17(2),

On the other hand, since