Research Article | Open Access

# Generalizations of -Fuzzy Filters in -Algebras

**Academic Editor:**Pentti Haukkanen

#### Abstract

Generalizations of a part of the paper (Ma et al., 2009) are considered. As a generalization of an -fuzzy filter, the notion of an -fuzzy filter is introduced, and its characterizations are provided. The implication-based fuzzy filters of an -algebra are discussed.

#### 1. Introduction

One important task of artificial intelligence is to make the computers simulate beings in dealing with certainty and uncertainty in information. Logic appears in a “sacred” (resp., a “profane”) form which is dominant in proof theory (resp., model theory). The role of logic in mathematics and computer science is twofold—as a tool for applications in both areas, and a technique for laying the foundations. Nonclassical logic including many-valued logic and fuzzy logic takes the advantage of classical logic to handle information with various facets of uncertainty (see [1] for generalized theory of uncertainty), such as fuzziness and randomness. Nonclassical logic has become a formal and useful tool for computer science to deal with fuzzy information and uncertain information. Among all kinds of uncertainties, incomparability is an important one which can be encountered in our life. The concept of -algebras was first introduced by Wang in [2] by providing an algebraic proof of the completeness theorem of a formal deductive system [3]. Obviously, -algebras are different from the BL-algebras. Jun and Lianzhen [4] studied filters of -algebras. Lianzhen and Kaitai [5] discussed the fuzzy set theory of filters in -algebras. As a generalization of the notion of fuzzy filters, Ma et al. [6] dealt with the notion of -fuzzy filters in -algebras.

In this article, we try to get more general form of the notion of -fuzzy filters. We introduce the notion of -fuzzy filters and investigate related properties. We establish characterizations of an -fuzzy filter and finally consider the implication-based fuzzy filters of an -algebra. The important achievement of the study with an -fuzzy filter is that the notion of an -fuzzy filter is a special case of an -fuzzy filter, and thus the related results obtained in the paper [6] are a corollary of our results obtained in this paper.

#### 2. Preliminaries

*Definition 2.1 (see [2]). *Let be a bounded distributive lattice with order-reversing involution and a binary operation . Then is called an -algebra if it satisfies the following axioms: (R1)(R2)(R3)(R4)(R5)(R6)

Let be an -algebra. For any we define and It is proved that and are commutative, associative, and and is a residuated lattice.

*Example 2.2 (see [5]). *Let For any we define and
Then is an -algebra which is neither a -algebra nor a lattice implication algebra.

An -algebra has the following useful properties.

Proposition 2.3 (see [7]). *For any elements and of an -algebra one has the following properties: *(a1)* if and only if *(a2)(a3)(a4)(a5)* implies *(a6)* implies *(a7)(a8)(a9)* and *(a10)* and *(a11)(a12)(a13)* if and only if *(a14)* implies *(a15)(a16)

A nonempty subset of an -algebra is called a *filter* of if it satisfies the following two conditions: (b1)(b2)

It can be easily verified that a nonempty subset of an -algebra is a filter of if and only if it satisfies the following conditions: (b3)(b4)

*Definition 2.4. *A fuzzy set in an -algebra is called a *fuzzy filter* of if it satisfies the following: (c1)(c2) is order-preserving, that is,

Theorem 2.5. *A fuzzy set in an -algebra is a fuzzy filter of if and only if it satisfies the following: *(c3)(c4)

For any fuzzy set in and the set
is called a *level subset* of A fuzzy set in a set of the form
is said to be a *fuzzy point* with support and value and is denoted by

For a fuzzy point and a fuzzy set in a set Pu and Liu [8] introduced the symbol where To say that (resp. ), we mean (resp. ), and in this case, is said to *belong to* (resp. *be quasi-coincident with*) a fuzzy set To say that (resp. ), we mean that or (resp. and ).

#### 3. Generalizations of -Fuzzy Filters

In what follows, is an -algebra and let denote an arbitrary element of unless otherwise specified. To say that we mean To say that we mean or For to say that we mean does not hold.

*Definition 3.1. *A fuzzy set in is said to be an *-fuzzy filter* of if it satisfies the following: (d1)(d2) for all and

An -fuzzy filter of with is called an *-fuzzy filter* of

*Example 3.2. *Let be a set with Hasse diagram and Cayley tables which are given in Table 1. Then is an -algebra (see [5]), where and Define a fuzzy set in by
It is routine to verify that is an -fuzzy filter of But it is neither a fuzzy filter nor an -fuzzy filter of since and and but

Theorem 3.3. *Every fuzzy filter is an -fuzzy filter.*

*Proof. *It is straightforward.

Example 3.2 shows that the converse of Theorem 3.3 may not be true and shows that an -fuzzy filter may not be an -fuzzy filter in general.

We establish characterizations of an -fuzzy filter.

Theorem 3.4. *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following: *(d3)(d4)

*Proof. *Let be an -fuzzy filter of Assume that (d3) is not valid. Then there exist such that
If then Hence
for some It follows that and but Moreover, and so Consequently a contradiction. If then and Thus and but Also,
that is, Hence again, a contradiction. Therefore (d3) is valid. Let be such that Assume that Then
for some If then by (3.5). Hence and Furthermore, that is, Thus a contradiction. If then by (3.5). Hence and Also, that is, Thus which is also a contradiction. Therefore for all with that is, (d4) is valid.

Conversely, let be a fuzzy set in satisfying two conditions (d3) and (d4). Let and be such that and Then and It follows from (d3) that
The case implies that From the case we have
that is, Hence Finally let and be such that and Then and so
by (d4). If then and thus If then which implies that that is, Thus Consequently, is an -fuzzy filter of .

If we take in Theorem 3.4, then we have the following corollary.

Corollary 3.5 (see [6]). *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following: *(1)(2)

Theorem 3.6. *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following: *(d5)(d6)

*Proof. *Let be an -fuzzy filter of Since for all it follows from (d4) that for all Let Since we have by (a13). Using (d4) and (d3), we obtain

Conversely, let be a fuzzy set in satisfying two conditions (d5) and (d6). Let be such that Then and so
by (d6) and (d5). Note from (a11) that
for all It follows from (d5) and (d6) that
Using Theorem 3.4, we conclude that is an -fuzzy filter of

Corollary 3.7 (see [6]). *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following: *(1)(2)

*Proof. *It is straightforward by taking in Theorem 3.6.

Corollary 3.8. *If is an -fuzzy filter of with then is a fuzzy filter.*

If we take in Corollary 3.8, then we obtain the following corollary.

Corollary 3.9 (see [6]). *If is an -fuzzy filter of with then is a fuzzy filter.*

Theorem 3.10. *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following: *(d7)

*Proof. *Assume that is an -fuzzy filter of Let be such that It follows from (d4) that and so from (d6) that

Conversely, let be a fuzzy set in satisfying (d7). Since for all we have by (d7). Note that for all It follows from (d7) that
Using Theorem 3.6, we conclude that is an -fuzzy filter of

Corollary 3.11 (see [6]). *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following:
*

*Proof. *It is obvious by taking in Theorem 3.10.

Theorem 3.12. *For an -fuzzy filter of the followings are equivalent: *(1)(2)(3)

*Proof. *()(): Suppose that satisfies the condition (1). If we take and in (1), then
for all by (d5).

()(): Assume that satisfies the condition (2). Note that
for all It follows from (R4), (2), and (d4) that
for all .

()(): Suppose that satisfies the condition (3). Using (d6), we have
for all

Corollary 3.13 (see [6]). *For an -fuzzy filter of the followings are equivalent: *(1)*,*(2)(3)

Theorem 3.14. * A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following:
*

*Proof. *Assume that is an -fuzzy filter of Let be such that Let Then and Hence by (d3), and so Let be such that If then
by (d4). This shows that Therefore is a filter of

Conversely, let be a fuzzy set in satisfying (3.20). If there exist such that then
for some Thus and which is a contradiction. Hence for all Let be such that If then
for some It follows that and a contradiction. Therefore for all with Using Theorem 3.4, we conclude that is an -fuzzy filter of .

Corollary 3.15 (see [6]). *A fuzzy set in is an -fuzzy filter of if and only if it satisfies the following:
*

Theorem 3.16. *If is a filter of then a fuzzy set in such that for and otherwise, where and is an -fuzzy filter of *

*Proof. *Note that
which is a filter of It follows from Theorem 3.14 that is an -fuzzy filter of

Corollary 3.17. *If is a filter of then a fuzzy set in such that for and otherwise, where and is an -fuzzy filter of *

For any fuzzy set in and any we consider four subsets:

It is clear that and

Theorem 3.18. *If is an -fuzzy filter of then
*

*Proof. *Assume that is an -fuzzy filter of and let be such that Let Then and that is, and Using (d3), we have
that is, Hence Let be such that If then that is, It follows from (d4) that
that is, Hence Therefore is a filter of

Corollary 3.19. *If is an -fuzzy filter of then
*

*Proof. *It is clear by taking in Theorem 3.18.

Theorem 3.20. *For any fuzzy set in the followings are equivalent: *(1)* is an -fuzzy filter of *(2)* is a filter of *

*Proof. *Assume that is an -fuzzy filter of and let be such that Let Then or and or We can consider four cases:
For the first case, (d3) implies that
and so that is, or Therefore For the case (3.31), assume that Then and so
Thus Suppose that Then which implies that
and thus We have similar result for the case (3.32). For the final case, if then Hence
and so If then
which implies that Let be such that If then or Assume that (d4) implies that
so that Suppose that If then
and thus If then
which implies that Therefore is a filter of

Conversely, let be a fuzzy set in such that is a filter of whenever it is nonempty for all If there exists such that then for some It follows that but Also, and so that is, Therefore a contradiction. Hence for all Suppose that there exist such that Then
for some which implies that and so from (b2) that since is a filter of But, (3.42) implies that and that is, This is a contradiction, and therefore for all Using Theorem 3.6, we conclude that is an -fuzzy filter of

If we take in Theorem 3.20, then we have the following corollary.

Corollary 3.21. *For any fuzzy set in the followings are equivalent: *(1)* is an -fuzzy filter of *(2)* is a filter of *

#### 4. Implication-Based Fuzzy Filters

Fuzzy logic is an extension of set theoretic multivalued logic in which the truth values are linguistic variables or terms of the linguistic variable truth. Some operators, for example, in fuzzy logic are also defined by using truth tables and the extension principle can be applied to derive definitions of the operators. In fuzzy logic, the truth value of fuzzy proposition is denoted by For a universe of discourse, we display the fuzzy logical and corresponding set-theoretical notations used in this paper: The truth valuation rules given in (4.3) are those in the ukasiewicz system of continuous-valued logic. Of course, various implication operators have been defined. We show only a selection of them in the following. (a)Gaines-Rescher implication operator () is (b)Gödel implication operator () is (c)The contraposition of Gödel implication operator () is

Ying [9] introduced the concept of fuzzifying topology. We can expand his/her idea to -algebras, and we define a fuzzifying filter as follows.

*Definition 4.1. *A fuzzy subset of is called a * fuzzifying filter* of if it satisfies the following conditions: (1)for all we have
(2)for all we get

Obviously, conditions (4.9) and (4.10) are equivalent to (c3) and (c4), respectively. Therefore a fuzzifying filter is an ordinary fuzzy filter.

In [10], the concept of -tautology is introduced, that is,

*Definition 4.2. *Let be a fuzzy set in and Then is called a *-implication-based fuzzy filter* of if it satisfies the following conditions: (1)for all we have
(2)for all we get