#### Abstract

The most simplified axiom systems of pseudo-weak- algebras and pseudo- algebras are obtained, and the mutually independence of axioms is proved. We introduce the notions of filters and normal filters in pseudo-weak- algebras. The structures and properties of the generated filters and generated normal filters in pseudo-weak- algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak- algebras.

#### 1. Introduction

In recent years, the study of logic algebras and their noncommutative generalizationâ€”pseudo-logic algebrasâ€”has become of greater focus in the field of logic. BCK and BCI algebras were introduced by Imai and Iseki [1] and have been extensively investigated by many researchers. Georgescu and Iorgulescu [2] introduced the notion of a pseudo-BCK algebra as a noncommutative generalization of a BCK-algebra. Liu et al. [3] investigated the theory of pseudo-BCK algebras. MV-algebras were introduced by Chang in [4] as an algebraic tool to study the infinitely valued logic of Lukasiewicz. Georgescu and Iorgulescu [5] introduced pseudo MV-algebras which is a noncommutative generalization of MV-algebras. The notion of BL-algebras was introduced by Hajek [6] as the algebraic structures for his Basic Logic. Georgescu and Iorgulescu [7] introduced the notion of pseudo-BL algebras by dropping commutative axioms in BL-algebras. di Nola et al. [8, 9], Zhang and Fan [10], and Zhan et al. [11] investigated in detail the theory of pseudo-BL algebras. MTL-algebras [12] are the algebraic structures for Esteva-Godo monoidal -norm based logic, many-valued propositional calculus that formalizes the structure of the real unit interval , induced by a left-continuous -norm. Flondor et al. [13] presented pseudo-MTL algebras as a noncommutative generalization of MTL-algebras. IMTL-algebras [12] are the algebraic counterpart for involutive monoidal -norm logic, an extension of MTL-algebras. NM-algebras [12] are the algebraic counterpart for nilpotent minimum logic, an extension of IMTL-algebras. Iorgulescu [14] and Liu and zhang [15] introduced and studied the pseudo-IMTL algebras and pseudo-NM algebras. algebras were introduced by Wang [16] as the algebraic structure for his formal deductive system of fuzzy propositional calculus. Weak- algebras [16] are the generalization of algebras. The research on algebras has attracted more and more attention [17].

In [18], we introduced and studied the pseudo-weak- algebras and pseudo- algebras. They are noncommutative generalizations of the weak- algebras and algebras, respectively. Some properties, the noncommutative forms of the properties in weak- algebras and algebras, were investigated. We showed that pseudo-weak- algebras are categorically isomorphic to pseudo-IMTL algebras, and pseudo- algebras are categorically isomorphic to pseudo-NM algebras.

Based on these results, in this paper, our study focused on the axioms independence and filter theory in pseudo-weak- algebras and pseudo- algebras. The most simplified axiom systems of pseudo-weak- algebras and pseudo- algebras are obtained, and the mutually independence of axioms is proved. The notions of filters and normal filters in pseudo-weak- algebras are introduced. The structures and properties of the generated filters and generated normal filters in pseudo-weak- algebras are obtained. These can be seen as noncommutative generalizations of the corresponding ones in weak- algebras.

#### 2. Preliminaries

We recall some definitions and results which will be used in the sequel.

*Definition 1 (see [12]). *An IMTL (involutive MTL) algebra is a structure of type such that for all :(B1) is a bounded lattice,(B2) is a monoid,(B3) if and only if ,(B4),(B5),where .

An NM (nilpotent minimum) algebra is an IMTL algebra satisfying the following condition:(B6).

*Definition 2 (see [14, 15]). *A pseudo-IMTL (pseudo-involutive MTL) algebra is a structure of type such that for all :(pB1) is a bounded lattice,(pB2) is a monoid,(pB3) if and only if if and only if ,(pB4),(pB5),where and .

A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying the following condition:(pB6).

*Definition 3 (see [16, 19]). *Let be a -type algebra, where is a unary operation and , , and are binary operations. If there is a partial ordering on , such that is a bounded distributive lattice, and are infimum and supremum operations with respect to , is an order-reversing involution with respect to , and the following conditions hold for any (R1),(R2), ,(R3),(R4),(R5), ,where 1 is the largest element of , and then we call a weak- algebra.

An algebra is a weak- algebra satisfying the additional condition as follows:(R6).

*Definition 4 (see [18]). *A pseudo-weak- algebra is a structure such that is a bounded distributive lattice, and are order-reversing pseudo-involution (i.e., if , then and ; ), and the following axioms hold for any :(pR1), ,(pR2*); ,(pR3), ,(pR4),(pR5*); .

A pseudo- algebra is a pseudo-weak- algebra satisfying the additional axiom as follows:(pR6).

In [18], we also have another simplified definition.

*Definition 5 (see [18]). *A pseudo-weak- algebra is a structure satisfying(pL1) is a bounded lattice,(pL2)if , then and ,(pL3),(pR1), ,(pR2),(pR3), ,(pR4),(pR5), .

A pseudo- algebra is a pseudo-weak- algebra satisfying the additional axiom as follows:(pR6).

Proposition 6 (see [18]). * In a pseudo-weak- algebra, the following properties hold:*(1)

*(2)*

*,**,**(3)*

*,**,**(4)*

*,**(5)*

*if and only if**if and only if**,**(6)*

*,**, whenever the arbitrary meets and unions exist,**(7)*

*,**, whenever the arbitrary meets and unions exist,**(8)*

*if**, then**and**,**(9)*

*if**, then**and**,**(10)*

*,**,**(11)*

*,**,**(12)*

*,**,**(13)*

*,**,**(14)*

*,**,**(15)*

*is a bounded distributive lattice,**(16)*

*,**,**(17)*

*,**,**(18)*

*,**,**(19)*

*,**(20)*

*,**,**(21)*

*,**,**(22)*

*,**(23)*

*,**(24)*

*,**(25)*

*,**(26)*

*,**,**(27)*

*,*,*(28)*

*if and only if**,**(29)*

*,*,

*.**(30)*

*In a pseudo-weak-**algebra (pseudo-**algebra)**, we define a binary operation**as follows, for any**:*Proposition 7 (see [18]). * In a pseudo-weak- algebra, the following properties hold:*(31)

*(32)*

*,**,**(33)*

*,**(34)*

*,**(35)*

*if and only if**if and only if**,**(36)*

*,**,**(37)*

*,**,**(38)*

*if**, then**and**,**(39)*

*,**,**(40)*

*,**(41)*

*,**, whenever the arbitrary unions exist,**(42)*

*,**,**(43)*

*,**, whenever the arbitrary meets exist,**(44)*

*,**, whenever the arbitrary unions and meets exist,**(45)*

*,**(46)*

*,**(47)*

*,**(48)*

*,**,*

*,**.*#### 3. The Axioms Independence of Pseudo-Weak- Algebras

We investigate the axioms independence of pseudo- algebras and pseudo-weak- algebras. Hence, we obtain most simplified axiom systems of pseudo-weak- algebras and pseudo- algebras.

Theorem 8. *A structure is a pseudo-weak- algebra if and only if it satisfies the following conditions:*(pL1)* is a bounded lattice,*(pL3â€˛)*, ,*(pR1)*, ,*(pR2)*,*(pR3)*, ,*(pR5)*, .*

*Proof. *Necessity is obvious. For sufficiency, it only needs to show axioms (pL2), (pL3), and (pR4) of Definition 5 hold. We first show the following three properties hold:(a), ,(b),(c) if and only if if and only if .

In fact, by (pR1) and (pR3), we have , .

By (a) and (pR2), we have , and so . Similarly, .

If , by (pR5) and (b), we have . Conversely, if , by (pR2) and (a), we have . Similarly, if and only if .

(pL2): by (c) and (pR1), if and only if if and only if if and only if . Similarly, if and only if if and only if if and only if .

(pL3): since , by (pL2), . By (pL3â€˛), ; thus and . Similarly, and .

By (pR2) and (pR1), , and so . Hence, . Similarly, we have .

(pR4): by (pL2) and (pL3), it is easy to verify that pseudo-Kleene dual law holds:(d), , , and .

By (pR1), (pR5), and (d), . Similarly, we have .

If , then and .

Now we prove that (pR4) holds. Since , . Hence, .

Corollary 9. *A structure is a pseudo- algebra if and only if it satisfies (pL1), (pL3â€˛), (pR1), (pR2), (pR3), (pR5), and*(pR6)*.**According to Theorem 8 and Corollary 9, one obtains most simplified definitions of pseudo-weak- algebras and pseudo- algebras, as the axiom systems are mutually independence (see Theorem 11).*

*Definition 10. *A pseudo-weak- algebra is a structure such that is a bounded lattice and and , satisfying the following axioms:(P1), ,(P2),(P3), ,(P4), .

A pseudo- algebra is a pseudo-weak- algebra satisfying the additional axiom as follows:(P5).

Theorem 11. *The five axioms of Definition 10 are mutually independent.*

*Proof. *Let , , , , and . Then is a bounded lattice satisfying for any .(i)Define operations and as pseudo-Godel implication on as follows:

Then satisfies (P2)â€“(P5), but not (P1): , but .(ii)Define operations and on as follows:

Clearly, satisfies (P1) and (P3)â€“(P5), but not (P2): .(iii)Define operations and on as follows:

Then satisfies (P1)-(P2) and (P4)-(P5), but not (P3). In fact, let , , and , then .(iv)Define operations and as pseudo-Lukasiewicz implication on as follows:

Then satisfies (P1)â€“(P4), but not (P5): .(v)Suppose that is a bounded lattice given by Figure 1.

The operations , , , and on are defined by the following:

Then satisfies (P1)â€“(P3) and (P5), but not (P4): , but .

#### 4. Filters and Normal Filters of Pseudo-Weak- Algebras

We introduce the notions of filters and normal filters in pseudo-weak- algebras and investigate the structures and properties of the generated filters and generated normal filters in pseudo-weak- algebras.

*Definition 12. *A nonempty subset of a pseudo-weak- algebra is said to be a filter of if it satisfies(F1),
(F2), .

Proposition 13. *For a subset of a pseudo-weak- algebra , the following are equivalent:*(i)* is a filter,*(ii)

*(iii)*

*and**,*

*and**.**Proof. *(i)(ii). By (F2), we have . By (F1), . By (38) and (F2), , and so .

(ii)(iii). If , by (19), . By (4), . By (ii), .

(iii)(i). If , , then , so ; that is, (F2) holds; if , by (41), , and so , which means (F1) holds.

Clearly, and are both filters of a pseudo-weak- algebra .

Proposition 14. * For a subset of a pseudo-weak- algebra , the following are equivalent:*(i)

*(ii)*

*is a filter,**(iii)*

*,**,*

*,**.**Proof. *(i)(ii). If , , by (F2) and Proposition 13 (iii), . Conversely, if , by , we have ; suppose that , by , we have . By Proposition 13 (iii), is a filter.

(i)(iii). Similarly.

Next, we consider filter generated by a set. It is easy to verify that the intersection of filters of is also a filter. If , the least filter containing ; that is, the intersection of all filters of containing is called the filter generated by and denoted by . If , is written . Clearly

Theorem 15. *Let be a pseudo-weak- algebra and let be a nonempty subset of . Then
*

*Proof. *Only prove the first equality. Using (34) to the first equality, we can get the rest of the two equalities. Let denote the right side of the first equality. If , then there are such that and . By (37), , so . If and , we have , so . Hence is a filter. If is a filter and , for any , there are such that . By (F2), , hence .

For convenience, we shall write and ; and ; and .

Corollary 16. *If is a pseudo-weak- algebra and , then
*

Corollary 17. *Let ** be a filter of a pseudo-weak-** algebra ** and **; then*

Theorem 18. *Let be a filter of a pseudo-weak- algebra and ; then
*

*Proof. *Assume that , by Corollary 17, there are such that
Put and , and then
Thus, by (46) . . Hence . Inverse contains is obvious.

Corollary 19. *Let be a filter of a pseudo-weak- algebra and . If , then
*

Corollary 20. *Let be a pseudo-weak- algebra and ; then .*

*Proof. *Taking in Theorem 18.

Next we introduce the notion of normal filters in a pseudo-weak- algebra.

*Definition 21. *A filter of a pseudo-weak- algebra is called normal if , if and only if .

Proposition 22. *Let be a normal filter of a pseudo-weak- algebra . Then there is such that if and only if there is such that .*

*Proof. *If there is such that , by (34), . By , we have , and so . Put then . Converse is similar.

Theorem 23. *If ** is a normal filter of a pseudo-weak-** algebra ** and **, then*

*Proof. *We show the first equality. By Corollary 17,
Since
by (34),
by Proposition 22, there is such that
and so
Repeating the above steps, there are such that
Let and , we have . That is that the first equality holds.

By the first equality and Proposition 22, we can obtain the second equation.

Corollary 24. *If ** is a normal filter of a pseudo-weak-** algebra ** and **, then*

*Proof. *By Theorem 23,

Since there is such that , if and only if there is such that ; that is, there is such that , if and only if . Thus, we prove the first equality.

Similarly, by
Since there is such that , if and only if there is such that ; that is, there is such that , if and only if , Thus, we have the second equality.

Corollary 25. *If ** is a normal filter of a pseudo-weak-** algebra ** and **, then*

*Proof. *There is such that , if and only if there is such that , if and only if . There is such that , if and only if there is such that , if and only if . By Theorem 23, Corollary 25 holds.

#### 5. Conclusions

We obtained the most simplified axiom systems of pseudo-weak- algebras and pseudo- algebras and proved the mutually independence of axioms. We introduced the notions of filters and normal filters in pseudo-weak- algebras and gave the structures and properties of the generated filters and generated normal filters in pseudo-weak- algebras. These will be conducive to further study pseudo-weak- algebras (pseudo-IMTL algebras) and pseudo- algebras (pseudo-NM algebras). In the future, we will investigate relations between various kinds of filters of pseudo-logic algebras. We may also study fuzzy type of filters of pseudo-weak- algebras and pseudo- algebras.

#### Conflict of Interests

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

#### Acknowledgments

This work is supported by the National Natural Science Foundations of China (61175055), the Fujian Province Natural Science Foundations of China (2013J01017), and the Fujian Province Key Project of Science and Technology of China (2011Y0049).