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Mathematical Problems in Engineering
Volume 2013, Article ID 908623, 6 pages
http://dx.doi.org/10.1155/2013/908623
Research Article

A Bijection between Lattice-Valued Filters and Lattice-Valued Congruences in Residuated Lattices

1School of Computer Science and Engineering, Xi’an University of Technology, Xi’an, Shaanxi 710048, China
2College of Computer Science and Technology, Taiyuan University of Technology, Taiyuan 030024, China

Received 22 January 2013; Accepted 1 July 2013

Academic Editor: Bin Liu

Copyright © 2013 Wei Wei et al. 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 study relations between lattice-valued filters and lattice-valued congruences in residuated lattices. We introduce a new definition of congruences which just depends on the meet and the residuum . Then it is shown that each of these congruences is automatically a universal-algebra-congruence. Also, lattice-valued filters and lattice-valued congruences are studied, and it is shown that there is a one-to-one correspondence between the set of all (lattice-valued) filters and the set of all (lattice-valued) congruences.

1. Introduction and Preliminaries

The interest in lattice-valued logic has been rapidly growing recently. Several algebras playing the role of structures of true values have been introduced and axiomatized [13]. The most general structure considered in this paper is that of a residuated lattice [4].

In a narrow sense, a residuated lattice is an algebra of type satisfying the following: (i) is a bounded lattice with as the bottom element, and the top element respectively; (ii) is a commutative monoid and monotone at both arguments; (iii) if and only if . The operations are called the multiplication and residuum, respectively. A residuated lattice in this paper is generally called a bounded, integral, and commutative residuated lattice in [4].

Residuated lattices were first introduced as a generalization of ideal lattices of rings in 1939 by Ward and Dilworth [5]. In their original definition, a residuated lattice was what we would call an integral commutative one.

For a residuated lattice , the negation operation is defined by .

Residuated lattices are very common in mathematical science and a lot of lattices and algebras are residuated lattices firstly. For example, an integral commutative Girard-monoid [2] is a residuated lattice satisfying the law of double negation: ; a Heyting algebra [6] is a residuated lattice with ; an MV-algebra [7] is a residuated lattice where holds; an MTL-algebra [1] is a residuated lattice satisfying ; a BL-algebra [8] is an MTL-algebra satisfying ; a product algebra (or -algebra) [8] is a BL-algebra satisfying and ; a -algebra (Gödel algebra) [2] is both a Heyting algebra and an MTL-algebra; an -algebra [3] is a residuated lattice where ; a lattice implication algebra [9] is a residuated lattice with (where : is an order-reversing involution).

Since the class of all residuated lattices is a variety of algebras (Proposition  2 in [10]), we can study them as universal algebras. Now, consider a residuated lattice as a universal algebra; a congruence on is an equivalence relation which preserves all operators on ; that is, implies that .

The aim of this paper is to study the relation between lattice-valued filters and lattice-valued congruences in residuated lattices. We will introduce a new definition of congruences just depending on the meet and the residuum . Then it is shown that each of these congruences is automatically a universal-algebra-congruence. Also, lattice-valued filters and lattice-valued congruences are studied, and it is shown that there is a one-to-one correspondence between the set of all (lattice-valued) filters and the set of all (lattice-valued) congruences.

2. Filters and Congruences

In pure mathematics, (lattice-valued) filters (or ideals) and (lattice-valued) congruences are useful tools in investigating the structure of the corresponding algebras.

The definition of a residuated lattice (in a narrow sense) has been given in Section 1. In the following discussion, always denotes a residuated lattice.

Proposition 1 (see [3, 8, 10, 11]). Let be a residuated lattice. Then(R1);(R2);(R3);(R4);(R5);(R6);(R7)  if and only if  ;(R8)  if and only if  ;(R9);(R10);(R11);(R12);(R13);(R14);(R15);(R16).

Definition 2 (see [8]). A nonempty subset of is called a filter if(F1) is an upper set; that is, and imply for all ;(F2) is closed under ; that is, holds for all .

Proposition 3 (see [8]). Let be a nonempty subset of  . Then the following three are equivalent:(1)  is a filter; (2)  and    imply    for all  ;(3)  is closed under    and    for all    and  .

Denote as the set of all filters of  . Then is a complete lattice under the partial order of set inclusion with the largest element and the least element . Furthermore, the meets in are the usual intersection of sets.

For simplification of congruence relation in algebraic structures, related attempts have been made in [1214].

Definition 4. A nonempty subset of is called a -congruence on if the following conditions hold:(ER) is an equivalence;
for any ,(C1) if , then ;(C2) if , then .
Obviously, a congruence is always a -congruence. Let denote the set of all congruences on . It is easy to verify that is a complete lattice, where the meets are the usual intersection of sets and , are the largest and the least elements, respectively.
Let be a congruence on and , where is the congruence class of with respect to . Define , , , . It is easy to verify that , is also a residuated lattice.

Proposition 5 (see [15]). Let be a filter of . Then is a -congruence on .

Proof. (ER) Obviously, is reflexive and symmetric. To show the transitivity of , suppose that ; we have . Then By (R16) and is an upper set; we have .
Suppose that and . Then .
(C1) First, Then since is an upper set. Similarly, we have . Hence .
(C2) By (R16), we have Thus since is an upper set. Similarly, we have . Hence .

Proposition 6. Let be a filter of . If , then ,.

Proof. Suppose that . Then .
(1) . In fact, by (R1) and (R2), It follows that . Similarly, . Hence .
(2) . In fact, by (R1) and (R3), It follows that . Similarly, . Hence .
(3) . In fact, by (R10), , which implies that . Similarly, . Thus . Similarly, . Hence by the transitivity of .
(4) . In fact, by (R16), we have which implies that Thus, . Similarly, . Hence .

Proposition 7. Let     be a -congruence on . Then is a filter of .

Proof. Obviously, . Suppose that ; that is, . By (R6) and (C2), we have and by the transitivity of  , we have . Thus . Hence is a filter of .

Lemma 8. Let be a -congruence on . Then if and only if and .

Proof. Suppose that . Then and similarly . Conversely, suppose that and . Then By (C1) and (R15), Similarly, we have . Hence by the transitivity of .

Theorem 9. Let be a filter of and a -congruence on , respectively. Then and . Thus there is a bijection between and .

Proof. (1) By Lemma 8, if and only if and if and only if and if and only if . Hence .
(2) if and only if if and only if and if and only if . Hence .

Remark 10. (1) By Proposition 6 and Theorem 9, if is a -congruence on and , then . That is to say, a -congruence and a (universal) congruence are equivalent to each other, and so are the symbols .
(2) In [16], Pavelka firstly showed that there is a one-to-one correspondence between all filters and all congruences in a residuated lattice. And a binary relation is a universal-algebra-congruence if and only if it is an equivalence relation that preserves both and (that is, it just depends on the operations ; the other two operations are automatically preserved).

3. -Filters

In the following part of this paper, unless otherwise stated, always denotes a lattice with a greatest element 1. In a lattice , an element is called prime (resp., coprime) if (resp., ) always implies or (resp., or ) for all . The set of all prime (resp., coprime) elements of is denoted by (resp., ). A complete lattice is called a spatial frame [6] if and is called a closed set lattice [17] if .

In this section, we will study -filters and their properties in the residuated lattice .

Definition 11. We call a mapping a lattice-valued filter of if(FF1) ;(FF2) for all .

Remark 12. The definition of a lattice-valued filter [13] is a lattice-valued set of satisfying (FF2) and() for all  ,which is different from Definition 11. It is easy to see that a lattice-valued filter in this paper is always a lattice-valued filter in [13]. In a common sense, a lattice-valued filter should be equivalent to a crisp one if we replaced by . Thus, the lattice-valued filter in [13] is not a direct generalization of a crisp one since (the constant map valued at 0) is a lattice-valued filter of while (the crisp counterpart) is not a crisp one.

Denote as the set of all lattice-valued filters of .

Proposition 13. Let be a mapping with . The following two are equivalent:(1);(2)  is monotone with respect to the order on    and     for all  .

Proof. : for any with , we have and Thus is monotone. By (FF2) and (R13), since is monotone. Also, since is monotone. Therefore, .
: by (R12), .

Corollary 14. If in , then for each , .

Proof. By Proposition 13 and (R1), . Then .

Let be a mapping. For any , define

Proposition 15. if and only if for any .

Proof. : clearly, for any . If , then . Then . Thus, . Hence .
: clearly, since . For any , suppose that . Then . Thus and . Hence .

Proposition 16. (1) If is a closed set lattice, then if and only if for any .
(2) If is a spatial frame, then if and only if for any .

Proof. (1) The necessity is from Proposition 15. Sufficiency: clearly, since for any . For any , suppose that and . Then . Thus and , By the arbitrariness of and .
(2) Necessity: clearly, for any since . If , then and . Then and . Hence . Sufficiency: if , then there exists such that . Then , which contradicts . Thus . For any , for any such that , we have and and then , which implies that and . By the arbitrariness of , we have . Hence .

4. Lattice-Valued Congruences

In this section, we will study lattice-valued congruences and the relations among filters, congruences, lattice-valued filters, and lattice-valued congruences in residuated lattices.

Definition 17. A mapping is called a lattice-valued congruence on if it satisfies the following, for any :(FC1) ;(FC2) ;(FC3) ;(FC4) ;(FC5) .

Denote as the set of all lattice-valued congruences on .

Definition 18. Let be a lattice-valued congruence on . Define by . is called the lattice-valued congruence class of with respect to on .

Proposition 19. Let be a lattice-valued congruence on . Then is a lattice-valued filter on , called the lattice-valued filter induced by , denoted by .

Proof. (FF1) Clearly, . (FF2) , by (FC3), and by (FC5), Thus . Hence .

Proposition 20. Let be a lattice-valued filter on and . Then is a lattice-valued congruence on , called the lattice-valued congruence induced by .

Proof. (FC1) and (FC2) are obvious and omitted. For any , (FC3) by Proposition 13 and (R16), (FC4) by Proposition 13, (R2), and (R7), (FC5) by Proposition 13, (R9), and (R15),

Theorem 21. Let be a lattice-valued congruence and a lattice-valued filter on , respectively. Then(1);(2).
Thus there is a bijection between and .

Proof. (1) , by (FC2)–(FC5), (R6), and (R15), And by (FC5) and (R7),
(2) For all  .

Lemma 22. Let be a lattice-valued congruence on . Then(1)for any , one has ;(2)if is a spatial frame, then for any , .

Proof. This proof is trivial by the definitions of congruences and lattice-valued congruences.

Proposition 23. For any , one has ,(1); (2).

Proof. (1) Consider the following:
(2) Consider

Replacing by in Proposition 23, we have the following.

Theorem 24. Let be a spatial frame. Then , and one has for any ,(1); (2).

By Theorem 9, Proposition 16, Theorem 21, Proposition 23 and Theorem 24, we have the following.

Corollary 25. (1) if and only if for any .
(2) if and only if for any .

By Corollary 25 and Remark 10, we have the following.

Corollary 26. Let be a lattice-valued congruence on . Then each of , and is larger than or equal to .

At last, we will give some properties of lattice-valued congruence classes of lattice-valued congruences.

Lemma 27. Let be a lattice-valued congruence on . Then for any .

Proof. It is a corollary of Theorem 21.

Proposition 28. Let be a lattice-valued congruence on and . Then the following four are equivalent.(1).(2).(3).(4).

Proof. Clearly, is equivalent to by Lemma 27. : . : for all  .Similarly, and so .
Similar to , we can show that .

Acknowledgments

The authors are thankful to Dr. Wei Yao and the anonymous reviewers for their valuable comments and suggestion. This work is supported by Scientific Research Program funded by Education Department of Shaanxi Province (no. 2013JK1139), National Natural Science Foundation of China (nos. 61202163 and 61240035) and Natural Science Foundation of Shaanxi Province (no. 2012011015-1).

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