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International Journal of Mathematics and Mathematical Sciences
Volume 2012 (2012), Article ID 274783, 11 pages
http://dx.doi.org/10.1155/2012/274783
Research Article

Pseudovaluations on WFI Algebras

1Department of Mathematics Education (and RINS), Gyeongsang National University, Chinju 660-701, Republic of Korea
2Department of Mathematics, Hanyang University, Seoul 133-791, Republic of Korea
3Department of Mathematics Education, Chinju National University of Education, Chinju 660-756, Republic of Korea

Received 24 August 2011; Accepted 21 October 2011

Academic Editor: Hee Sik Kim

Copyright © 2012 Young Bae Jun 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

Using Buşneag's model, the notion of pseudovaluations (valuations) on a WFI algebra is introduced, and a pseudometric is induced by a pseudovaluation on WFI algebras. Given a valuation with additional condition, we show that the binary operation in WFI algebras is uniformly continuous.

1. Introduction

In 1990, Wu [1] introduced the notion of fuzzy implication algebras (FI algebra, for short) and investigated several properties. In [2], Li and Zheng introduced the notion of distributive (regular, and commutative, resp.) FI algebras and investigated the relations between such FI algebras and MV algebras. In [3], Jun discussed several aspects of WFI algebras. He introduced the notion of associative (normal and medial, resp.) WFI algebras and investigated several properties. He gave conditions for a WFI algebra to be associative/medial, provided characterizations of associative/medial WFI algebras, and showed that every associative WFI algebra is a group in which every element is an involution. He also verified that the class of all medial WFI algebras is a variety. Jun et al. [4] introduced the concept of ideals of WFI algebras, and gave relations between a filter and an ideal. Moreover, they provided characterizations of an ideal, and established an extension property for an ideal. Buşneag [5] defined pseudovaluation on a Hilbert algebra and proved that every pseudovaluation induces a pseudometric on a Hilbert algebra. Also, Buşneag [6] provided several theorems on extensions of pseudovaluations. Buşneag [7] introduced the notions of pseudovaluations (valuations) on residuated lattices, and proved some theorems of extension for these (using the model of Hilbert algebras ([6])).

In this paper, using Buşneag’s model, we introduce the notion of pseudovaluations (valuations) on WFI algebras, and we induce a pseudometric by using a pseudovaluation on WFI algebras. Given a valuation with additional condition, we show that the binary operation in WFI algebras is uniformly continuous.

2. Preliminaries

Let be the class of all algebras of type . By a WFI algebra, we mean an algebra in which the following axioms hold:(a1),(a2),(a3),(a4).

For the convenience of notation, we will write for We define , and for , where occurs -times.

Proposition 2.1 (see [3]). In a WFI algebra , the following are true:(b1),(b2),(b3),(b4),(b5),(b6).

We define a relation “’’ on by if and only if . It is easy to verify that a WFI algebra is a partially ordered set with respect to . A nonempty subset of a WFI algebra is called a subalgebra of if whenever . A nonempty subset of a WFI algebra is called a filter of if it satisfies:(c1),(c2).

A filter of a WFI algebra is said to be closed (see [3]) if is also a subalgebra of . A nonempty subset of a WFI algebra is called an ideal of (see [4]) if it satisfies the condition (c1) and(c3).

Proposition 2.2 (see [3]). Let be a filter of a WFI algebra . Then is closed if and only if for all .

Proposition 2.3 (see [3]). In a finite WFI algebra, every filter is closed.

Note that every ideal of a WFI algebra is a closed filter (see [4, Theorem  4.3]). For a WFI algebra , the set is called the simulative part of .

3. WFI Algebras with Pseudovaluations

In what follows, let denote a WFI algebra unless otherwise specified.

Definition 3.1. A mapping is called a pesudovaluation on if it satisfies the following two conditions:(i),(ii).

A pseudovaluation on satisfying the following condition: is called a valuation on .

Obviously, a mapping is a pseudovaluation on , which is called the trivial pseudovaluation.

Example 3.2. Let be a mapping defined by where is a positive real number. Then, is a pseudovaluation on . Moreover, it is a valuation on .

Example 3.3. Let be the set of integers. Then, is a WFI algebra, where and for all (see [8]). Let be a mapping defined by for all , where and are real numbers with and . Then, is a pseudovaluation on .

Example 3.4. Let be a set with the following Cayley table: 274783.fig.001(3.5) Then, is a WFI algebra (see [3]). Define a mapping by and . Then, is a pseudovaluation on . Also, it is a valuation on .

Proposition 3.5. Every pseudovaluation on satisfies the following conditions:(1),(2),(3).

Proof. (1) Let be such that . Then, , and so
(2) Using (a4), we have for all . It follows from (1) and Definition 3.1(ii) that so that for all .
(3) Let . Using Definition 3.1(ii), we have and ; that is, and . It follows that .

Corollary 3.6. Let be a pseudovaluation on . Then, for all .

Proof. Since for all , we have by Proposition 3.5(1) and Definition 3.1(i).

The following example shows that the converse of Corollary 3.6 may not be true.

Example 3.7. Let be a WFI algebra which is considered in Example 3.4. Let be a mapping defined by Then, and . But is not a pseudovaluation on , since

Let be a pseudovaluation on . If , that is, , for all , then by Proposition 3.5(1). If for all , then , and so, by Proposition 3.5(1) and Definition 3.1(ii). Hence, . Now, if for all , then . It follows from Proposition 3.5(1) and Definition 3.1(ii) that so that . Continuing this process, we have the following proposition.

Proposition 3.8. Let be a pseudovaluation on . For any elements of , if , then .

Theorem 3.9. Let be a filter of , and let be a mapping defined by where is a positive real number. Then, is a pseudovaluation on . In particular, is a valuation on if and only if .

Proof. Straightforward.

We say is a pseudovaluation induced by a filter .

Theorem 3.10. If a mapping is a pseudovaluation on , then the set is a filter of .

Proof. Obviously, . Let be such that and . Then, and . It follows from Definition 3.1(ii) that so that . Hence, is a filter of .

We say is a filter induced by a pseudovaluation on .

Corollary 3.11. If a mapping is a pseudovaluation on a finite WFI algebra , then the set is a closed filter of .

Proof. It follows from Proposition 2.3 and Theorem 3.10.

Remark 3.12. A filter induced by a pseudovaluation on may not be closed. Indeed, in Example 3.3, if we take and , then , is a pseudovaluation on . Then, which is a filter but not a subalgebra of , since . Hence, is not a closed filter of .

Theorem 3.13. For any pseudovaluation , if is a filter of , then .

Proof. We have .

The following example shows that the converse of Theorem 3.10 may not be true; that is, there exist a WFI algebra and a mapping such that(1) is not a pseudovaluation on ,(2) is a filter of .

Example 3.14. Let be a set with the following Cayley table: 274783.fig.002(3.14) Then is a WFI algebra. Let be a mapping defined by Then, is a filter of . But is not a pseudovaluation on , since

Definition 3.15. A pseudovaluation (or, valuation) on is said to be positive if for all .

The pseudovaluation on which is given in Example 3.4 is positive.

Theorem 3.16. If a pseudovaluation on is positive, then the set is a filter of .

Proof. Clearly, . Let be such that and . Then, and . Since is positive, it follows from Definition 3.1(ii) that so that , that is, . Hence, is a filter of .

The following example shows that two distinct pseudovaluations induce the same filter.

Example 3.17. Consider a WFI algebra which is given in Example 3.14. Let and be mappings from to defined by Then, and are pseudovaluations on , and .

For a mapping , define a mapping by for all . Note that for all .

Theorem 3.18. If is a pseudovaluation on , then is a pseudometric on , and so is a pseudometric space.

We say is called the pseudometric induced by pseudovaluation .

Proof. Let . Then, by Proposition 3.5(3), and obviously, and . Now, Therefore, is a pseudometric space.

Proposition 3.19. Every pseudometric induced by pseudovaluation satisfies the following inequalities:(1), (2), (3), for all .

Proof. (1) Let . Since and , it follows from Proposition 3.5(1) that and so that
(2) It is similar to the proof of (1).
(3) Using Proposition 3.5(2), we have for all . Hence, for all .

Theorem 3.20. Let be a pseudovaluation on such that is a closed filter of . If is a metric on , then is a valuation on .

Proof. Suppose that is not a valuation on . Then, there exists such that and . Thus and so , since is a closed filter of . It follows that so that Hence, , and thus . Thus, since is a metric on . This is a contradiction. Therefore, is a valuation on .

Consider the pseudovaluation on which is described in Example 3.3. If , then for all , and which is not a closed filter of . Since is a pseudovaluation on , we know that is a pseudometric space by Theorem 3.18. If in , then Hence, is a metric space. But , and so, is not a valuation on . This shows that Theorem 3.20 may not be true when is not a closed filter of .

Theorem 3.21. For a mapping , if is a pseudometric on , then is a pseudometric space, where for all .

Proof. Suppose is a pseudometric on . For any , we have Now, let . Then, Therefore, is a pseudometric space.

Corollary 3.22. If is a pseudovaluation on , then is a pseudometric space.

It is natural to ask that if is a valuation on , then is a metric space. But, we see that it is incorrect in the following example.

Example 3.23. For a WFI algebra , a mapping defined by for all is a valuation on . Then, is a pseudometric on . Note that , but . Hence, is not a metric space.

Theorem 3.24. If is a positive valuation on , then is a metric space.

Proof. Suppose that is a positive valuation on . Then, is a pseudometric space by Theorem 3.18. Let be such that . Then, , and so and , since is positive. Also, since is a valuation on , it follows that and so from (a2) that . Therefore, is a metric space.

Corollary 3.25. If is a valuation on such that , then is a metric space.

Theorem 3.26. If is a positive valuation on , then is a metric space.

Proof. Note from Corollary 3.22 that is a pseudometric space. Let be such that . Then, and so , since for all . Hence, Since is positive, it follows that and so that and . Using (a2), we have and , and so . Therefore, is a metric space.

Theorem 3.27. If is a positive valuation on , then the operation is uniformly continuous. (Suppose that and are metric spaces and . We say that is uniformly continuous provided that for every , there exists such that for any points and in , if , then .)

Proof. For any , if , then , and . Using Proposition 3.19, we have Therefore, the operation is uniformly continuous.

Corollary 3.28. If is a valuation on such that , then the operation is uniformly continuous.

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