#### Abstract

Some new properties of fuzzy associative filters (also known as fuzzy associative pseudo-filters), fuzzy *p*-filter (also known as fuzzy pseudo-*p*-filters), and fuzzy *a*-filter (also known as fuzzy pseudo-*a*-filters) in pseudo-*BCI* algebras are investigated. By these properties, the following important results are proved: (1) a fuzzy filter (also known as fuzzy pseudo-filters) of a pseudo-*BCI* algebra is a fuzzy associative filter if and only if it is a fuzzy *a*-filter; (2) a filter (also known as pseudo-filter) of a pseudo-*BCI* algebra is associative if and only if it is an *a*-filter (also call it pseudo- filter); (3) a fuzzy filter of a pseudo-*BCI* algebra is fuzzy *a*-filter if and only if it is both a fuzzy *p*-filter and a fuzzy *q*-filter.

#### 1. Introduction and Preliminaries

In the field of artificial intelligence research, nonclassical logics (fuzzy logic, epistemic logic, nonmonotonic logic, default logic, etc.) are extensively used (see [1]). In this paper we discuss a kind of logic algebra system, that is, pseudo- algebra, which originated from -logic; it is a kind of nonclassical logic and inspired by the calculus of combinators [2]. Newly, in [3], we show that pseudo- algebra plays an important role in weakly integral residuated algebraic structure, which is in close connection with various fuzzy logic formal systems [4, 5].

In 1966, Iséki [2] introduced the concept of -algebra as an algebraic counterpart of the -logic. Since then, the ideal theory of -algebras gets in-depth research and development. In 2008, as a generalization of -algebra, Dudek and Jun [6] introduced the notion of pseudo- algebra which is also generalization of pseudo- algebra introduced by Georgescu and Iorgulescu in [7]. We investigated some classes of pseudo- algebras in [8]. Recently, the pseudoideal theory of pseudo- algebras has been studied: the notion of pseudo- ideal (or pseudoideal) of pseudo- algebra is introduced in [9]; some special pseudo- ideals are discussed in [10], for example, associative pseudoideal and pseudo- ideal, which are generalization of associative ideal of -algebra (it is introduced by X.H. Zhang and R.G. Ling).

The notion of fuzzy sets has been applied to many algebraic systems (see [4, 11, 12]); naturally, it has been applied to pseudo- algebra; for example, fuzzy pseudoideals have been investigated in [13–15]. As continuums of the above works, we further study fuzzy associative pseudoideal and fuzzy pseudo- ideal in pseudo- algebras.

Note that the notion of pseudo- algebra in this paper is indeed dual form of original definition in [6]; accordingly, the notion of pseudo-filter (or pseudo- filter) is the dual form of pseudoideal (or pseudo- ideal) in [9]. Moreover, for short, the notion of pseudo-filter (or pseudo- filter) is simply called “filter” in this paper.

At first, we recall some basic concepts and properties of pseudo- algebras.

*Definition 1 (see [6]). *A pseudo- algebra is a structure , where “” is a binary relation on , “” and “” are binary operations on , and “1” is an element of , verifying the axioms: for all ,(1), ;(2), ;(3);(4), ;(5).

If is a pseudo- algebra satisfying for all , then is a -algebra.

Proposition 2 (see [6, 9, 10]). *Let be a pseudo- algebra; then satisfy the following properties :*(1)*;*(2)*, **;*(3)*, **;*(4)*;*(5)*;*(6)*, **;*(7)*, **;*(8)*, **;*(9)*, **;*(10)*, **;*(11)*, **;*(12)*.*

*Definition 3 (see [9]). *A nonempty subset of a pseudo- algebra is called a pseudo- filter (briefly, filter) of if it satisfies(F1);(F2), ;(F3), .

*Definition 4 (see [10]). *A nonempty subset of a pseudo- algebra is called an associative pseudo- filter (briefly, associative filter) of if it satisfies(1);(2) and ;(3) and .

*Definition 5 (see [10]). *A nonempty subset of a pseudo- algebra is called a pseudo--filter (briefly, -filter) of if it satisfies(1);(2) and ;(3) and .

*Definition 6 (see [13–15]). *A fuzzy set is called a fuzzy pseudofilter (briefly, fuzzy filter) of pseudo- algebra if it satisfies(FF1), ;(FF2), ;(FF3), .

Proposition 7. *Let be a fuzzy filter of a pseudo- algebra . If , then , where .*

As a consequence of the so-called* Transfer Principle for Fuzzy Sets* in [11], we have the following.

Theorem 8 (see [11, 15]). *Let be a pseudo- algebra. Then a fuzzy set is a fuzzy filter of if and only if the level set is filter of for all .*

Theorem 9 (see [15]). *Let be a pseudo- algebra. Then a fuzzy set is a fuzzy filter of if and only if it satisfies*(1)*for all , ;*(2)*for all , .*

*Definition 10 (see [13, 15]). *A fuzzy set is called a fuzzy -filter of a pseudo- algebra if it satisfies (FF1) and(FPF1)for all , ;(FPF2)for all , .

*Definition 11 (see [13, 15]). *A fuzzy set is called a fuzzy -filter of a pseudo- algebra if it satisfies (FF1) and(FaF1)for all , ;(FaF2)for all , .

*Definition 12 (see [15]). *A fuzzy set is called a fuzzy associative filter of a pseudo- algebra if it satisfies (FF1) and(FAF1)for all , ;(FAF2)for all , .

*Definition 13 (see [13, 15]). *A fuzzy set is called a fuzzy -filter of a pseudo- algebra if it satisfies (FF1) and(FqF1)for all , ;(FqF2)for all , .

#### 2. New Properties of Fuzzy -Filters and Fuzzy Associative Filters

Lemma 14 (see [15]). *Let be a fuzzy -filter of a pseudo- algebra . Then satisfies
*

Lemma 15 (see [15]). *Let be a fuzzy -filter of a pseudo- algebra . Then satisfies
*

Theorem 16. *Let be a fuzzy -filter of a pseudo- algebra . Then the following statements hold for all :*(1)*for all , , ;*(2)*for all ;*(3)*for all , ;*(4)*for all , ;*(5)*for all , ;*(6)*for all , ;*(7)*for all , , .*

*Proof. *(1) For any , by Proposition 2(6), we have . From this, applying Proposition 7, we get
Using Definition 6(FF2), we have
Thus,
By Lemma 15, and . Therefore, .

Similarly, we have .

(2) For any , by Proposition 2(11) and (12), we have . By Lemma 14, it follows that
From this, applying Lemma 15, we get . Similarly, we can get . By Definition 1(4), .

(3) For any , applying Proposition 2(12) and Lemma 14, we have
By Definition 6(FF1), it follows that .

Similarly, we have .

(4) For any , by Proposition 2(6) and Definition 1(1), we have
From this, by Theorem 8, we get
Applying (3), Lemma 15, and Definition 6(FF1), . From this and (2), we get .

On the other hand, by Definition 1(1), . Using Proposition 7 and the above result, we have
Moreover, by (1), we have
Therefore,
This means that (4) holds.

(5) The proof is similar to (4).

(6) For any , by (4), we have
It follows that . Similarly, .

(7) For any , by Definition 1(3), we have . From this, (2), and Proposition 7, we get

Similarly, we have .

Lemma 17. *Let be a fuzzy associative filter of a pseudo- algebra . Then satisfies
*

*Proof. *It is easily proved by Definition 12; the proof is omitted.

Theorem 18. *Let be a fuzzy associative filter of a pseudo- algebra . Then the following statements hold:*(1)*for all , ;*(2)*for all , ;*(3)*for all , ;*(4)*for all , ;*(5)*for all , , ;*(6)*for all , , ;*(7)*for all , , ;*(8)*for all , , ;*(9)*for all , ;*(10)*for all , , ;*(11)*for all , ;*(12)*for all , , .*

*Proof. *(1) For any (by Definition 1 and Proposition 2),
From this and Lemma 17, we have
This means that . Similarly, .

(2) For any , by Definition 1(1), we have
Applying Theorem 9(1), we get
By (1) and Lemma 17, , . Therefore,

(3) It is similar to (2).

(4) For any , by (2), we have . On the other hand, applying Definition 1 and Proposition 2,
From this and Proposition 7, . Thus, . Similarly, we can get . Therefore, .

(5) For any , since (by Proposition 2)
then . By (1), ; hence, . Moreover, by Definition 6(FF3), we have
Thus,

Similarly, .

(6) By (2) and (3), we can get (6).

(7) For any , by Definition 1, we have .

Applying (1) and Theorem 9, we get

Similarly, .

(8) For any , by Lemma 17, . And, using Proposition 2(4), we have
Hence, . From this and (7), we get

Similarly, .

(9) By (1) and (8), we can get (8).

(10) It is similar to the proof of Theorem 16(7).

(11) For any , by Lemma 17, . On the other hand, using (8), . Hence, .

(12) It is similar to the proof of Theorem 16(1).

#### 3. Some Necessary and Sufficient Conditions for Fuzzy -Filters and Fuzzy Associative Filters

Checking the proof of Theorem 18 in detail, we know that the proof only applies the properties of fuzzy filters and the conditions in Lemma 17. From this, we can get the following.

Lemma 19. *Let be a fuzzy filter of a pseudo- algebra . If satisfies*(C1)*for all , ;*(C2)*for all , ,* *then the following statements hold:*(C3)*for all , ;*(C4)*for all , ;*(C5)*for all , ;*(C6)*for all , , .*

*Proof. *It is similar to Theorem 18 (the conditions (FAF1) and (FAF2) are not applied); the proof is omitted.

Theorem 20. *Let be a fuzzy filter of a pseudo- algebra . Then is a fuzzy associative filter of if and only if it satisfies*(C1)*for all , ;*(C2)*for all , .*

*Proof. *Assume that is a fuzzy associative filter of ; by Lemma 17, (C1) and (C2) hold.

Conversely, assume that satisfies conditions (C1) and (C2). For any , by Proposition 2(6), . Using Theorem 9, we get
By Lemma 19(C4) and Proposition 2(4), .

Thus,

(P1) .

On the other hand, applying Lemma 19(C5), Proposition 2(11), and (12),
And, by Lemma 19(C6),
By Lemma 19(C3) and the above result, we get

(P2) .

Combining (P1) and (P2), we get that . That is, (FAF1) holds. Similarly, condition (FAF2) holds. Therefore, by Definition 12, is a fuzzy associative filter of .

Theorem 21 (see [13, 15]). *Let be a fuzzy filter of a pseudo- algebra . Then is a fuzzy -filter of if and only if it satisfies*(a1)*for all , ;*(a2)*for all , .*

Theorem 22. *Let be a fuzzy filter of a pseudo- algebra . Then the following conditions are equivalent:*(i)* is a fuzzy -filter of ;*(ii)* is a fuzzy associative filter of .*

*Proof. *(i) (ii). Suppose that is a fuzzy -filter of . For any , by Definition 1(1),
Applying Theorem 16(6) and Proposition 7, we get
It follows that
Moreover, by Theorem 16(1),
Then we get
Using Theorem 16(2), . Thus,
On the other hand, applying Proposition 2(6) and (8), . From this and Theorem 16(1), we get
From , we have ; it follows that
Therefore,
This means that (C1) holds. Similarly, . By Theorem 20, is a fuzzy associative filter of .

(ii) (i). Suppose that is fuzzy associative filter of . For any , by Definition 1(1), . Applying Proposition 7, we get
On the other hand, using Theorem 18(12),
By Theorem 18(9), . It follows that
Therefore,

Similarly, we can get . By Theorem 21, is a fuzzy -filter of .

Now, we discuss the relationship between associative filters and -filters of pseudo- algebras. At first, we give the following results (the proofs are omitted).

Proposition 23. *A nonempty subset of pseudo- algebra is a filter (associative filter, -filter) of if and only if the characteristic function of is a fuzzy filter (fuzzy associative filter, fuzzy -filter) of .*

Proposition 24. *Let be a pseudo- algebra. Then a fuzzy set is a fuzzy associative filter (fuzzy -filter) of if and only if the level set is associative filter (-filter) of for all .*

In fact, the above proposition is a consequence of the so-called* Transfer Principle for Fuzzy Sets* in [11].

Combining Propositions 23 and 24, Theorems 8 and 22, we get the following.

Theorem 25. *Let be a filter of a pseudo- algebra . Then the following conditions are equivalent:*(i)* is an -filter of ;*(ii)* is an associative filter of .*

*Remark 26. *In [10], the authors proved Theorem 18, but the proof is wrong (from “Conversely …” to the end of proof). In fact, by Theorem 25, Theorem 4 in [10] is true.

Finally, we discuss the relationship among fuzzy associative filters, fuzzy -filters, and fuzzy -filters in pseudo- algebras.

Lemma 27. *Let be a fuzzy -filter of a pseudo- algebra . Then satisfies*(1)*for all , ;*(2)*for all , .*

*Proof. *(1) For any , by Definition 10(FPF1), we have
And, using Definition 1(2) and Proposition 7, . It follows that .

(2) For any , by Proposition 2(11), (12), and (9), we have
From this and (1), we get
This means that (2) holds.

Theorem 28. *Let be a fuzzy filter of a pseudo- algebra . Then the following conditions are equivalent:*(1)* is a fuzzy -filter of ;*(2)* is both a fuzzy -filter and a fuzzy -filter of .*

*Proof. *Assume that is a fuzzy -filter of . It is easy to prove that is both a fuzzy -filter and a fuzzy -filter of .

Conversely, let be both a fuzzy -filter and fuzzy -filter of . For any , by Definition 13(FqF1), we have
And, by Proposition 2(4),
From this and Lemma 27(2),
Therefore,
This means that Theorem 21(a1) holds. Similarly, we can prove (a2). By Theorem 21, we know that is a fuzzy -filter of .

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant no. 61175044) and Innovation Program of Shanghai Municipal Education Commission (no. 13ZZ122).