#### 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:
(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:
(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.*