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The Scientific World Journal
Volume 2014, Article ID 352381, 8 pages
http://dx.doi.org/10.1155/2014/352381
Research Article

Pseudo-Weak- Algebras

1Department of Mathematics and Computer, Wuyi University, Wuyishan, Fujian 354300, China
2Department of Computer Science and Technology, Chuxiong Normal University, Chuxiong, Yunnan 675000, China

Received 1 March 2014; Revised 7 April 2014; Accepted 10 April 2014; Published 4 May 2014

Academic Editor: Jianming Zhan

Copyright © 2014 Yong Lin Liu and Xiaobo Cai. 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

A positive answer to the open problem of Iorgulescu on extending weak- algebras and -algebras to the noncommutative forms is given. We show that pseudo-weak- algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo- algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak- algebras and -algebras, are investigated. The simplified axiom systems of pseudo-weak- algebras and pseudo- algebras are obtained.

1. Introduction

It is well known that certain information processing, especially inferences based on certain information, is based on classical two-valued logic. Due to strict and complete logical foundation (classical logic), making inferences about certain information can be done with high confidence levels. Thus, it is natural and necessary to attempt to establish some rational logic system as the logical foundation for uncertain information processing. It is evident that this kind of logic cannot be two-valued logic itself but might form a certain extension of two-valued logic. Various kinds of nonclassical logic systems have therefore been extensively researched in order to construct natural and efficient inference systems to deal with uncertainty.

In recent years, motivated by both theory and application, the study of t-norm-based logic systems and the corresponding pseudo-logic systems has become of greater focus in the field of logic. Here, t-norm-based logical investigations preceded the corresponding algebraic investigations, and in the case of pseudo-logic systems, algebraic development preceded the corresponding logical development.

A noncommutative generalization of reasoning can be found, for example, in psychological processes. In clinical medicine on behalf of transplantation of human organs, an experiment was performed in which the same two questions have been posed to two groups of interviewed people as follows. (1) Do you agree to donate your organs for medical transplantation after your death? (2) Do you agree to accept organs of a donor if you need them? When the order of questions was changed in the second group, the number of positive answers here was much higher than in the first group.

The following reviews some situation concerning some important logic algebras and the corresponding pseudo-logic algebras. 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 are 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 t-norm-based logic, a many-valued propositional calculus that formalizes the structure of the real unit interval , induced by a left-continuous t-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 t-norm logic, an extension of MTL-algebras. NM-algebras [12] are the algebraic counterpart for nilpotent minimum logic, an extension of IMTL-algebras. -algebras were introduced by Wang [14] as the algebraic structure for his formal deductive system of fuzzy propositional calculus. Weak--algebras [14] are the generalization of -algebras. In the recent years, the research on -algebras has attracted more and more attention [15]. In [16], Iorgulescu proposed one open problem (open problem 2.14 of [16]).

Problem. Recall that the IMTL-algebras, introduced in 2001 by Esteva and Godo, are categorically isomorphic to weak- algebras, introduced in 1997 by Wang, and that NM-algebras are categorically isomorphic to -algebras, introduced also in 1997 by Wang. Extend weak- algebras and -algebras to the noncommutative case.

In this paper, we extend weak- algebras and -algebras to the noncommutative forms, called pseudo-weak- algebras and pseudo- algebras. We show that pseudo-weak- algebras are categorically isomorphic to pseudo-IMTL-algebras and that pseudo- algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak- algebras and -algebras, are investigated. Furthermore, we discuss the simplified axiom systems of pseudo-weak- algebras and pseudo- algebras.

2. Preliminaries

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

Definition 1 (see [12]). An MTL-algebra (or a weak-BL-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).

Definition 2 (see [12]). An IMTL- (involutive MTL-) algebra is an MTL-algebra satisfying the following additional condition:(B5),where .
A WNM- (weak nilpotent minimum-) algebra is an MTL-algebra satisfying the following additional condition:(B6).
An NM- (nilpotent minimum-) algebra is an IMTL-algebra satisfying condition (B6).

Definition 3 (see [13]). A pseudo-MTL algebra (or a weak-pseudo-BL 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).

Definition 4 (see [16, 17]). A pseudo-IMTL (pseudo-involutive MTL) algebra is a pseudo-MTL algebra satisfying the following additional condition:(pB5),where and ,
A pseudo-WNM (pseudo-weak nilpotent minimum) algebra is a pseudo-MTL algebra satisfying the following additional condition:(pB6),where and .
A pseudo-NM (pseudo-nilpotent minimum) algebra is a pseudo-IMTL algebra satisfying condition (pB6).

Definition 5 (see [14, 18]). 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, 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 , then one calls a weak- algebra.
An algebra is a weak- algebra satisfying the following additional condition:(R6).

3. Pseudo-Weak- Algebras and Pseudo- Algebras

We introduce the notions of pseudo-weak- algebras and pseudo- algebras. They are noncommutative forms of weak- algebras and -algebras. Some of their properties are investigated.

Definition 6. 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) ,  ,() ; ,(pR3) , ,(pR4) ,() , ;, .
A pseudo- algebra is a pseudo-weak- algebra satisfying the following additional axiom:(pR6) .

Remark 7. (i) Comparing Definition 6 with Definition 5, the notions of pseudo-weak- algebras and pseudo- algebras are the noncommutative forms of weak- algebras and -algebras, respectively.
(ii) The operations and in Definition 6 are not primary (please see Proposition 10 (2)).
(iii) As we will see in Proposition 10 (3), (10), and (14), the following axioms in Definition 6 follow from the other axioms:(a)the distributivity of bounded lattice ,(b), (c), .Hence, we have the following simplified definition. In what follows, we will use the following definition.

Definition 8. A pseudo-weak- algebra is a structure satisfying the following:(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 following additional axiom:(pR6).

Example 9. Let , and define operations on as follows: Then, is a bounded lattice satisfying for any . Define operations and as pseudo-Lukasiewicz implication on : By routine calculations, is a pseudo-weak- algebra.

Proposition 10. In a pseudo-weak- algebra, the following properties hold:(1), ,(2), ,(3), (4) if and only if if and only if ,(5), , whenever the arbitrary meets and unions exist,(6), , whenever the arbitrary meets and unions exist,(7)if , then and ,(8)if , then and ,(9), ,(10),(11), ,(12), ,(13), ,(14) is a bounded distributive lattice.

Proof. (1) Since and , by (L3) and (L2), and , and hence . follows similarly.
(2) By (1), (pR1), and (pR2), , .
(3) By (pR1) and (pR2), . By (pR3) and (L3), ; that is, , and so . The second equality follows similarly.
(4) If , by (pR1), (pR3), and (pR2), . If , by (pR5) and (3), . Similarly, we have if and only if .
(5) Since for each , by (L2), for each , and so . Since for each , by (L2) and (L3), for each , and so , . The second equality follows similarly.
(6) Similarly.
(7) If , by (pR5), , and so . Similarly, .
(8) If , then . By (pR1), . Similarly, .
(9) By (pR1), (5), and (pR5), . Similarly, .
(10) By (4), (pR1), and (pR4), for any , if and only if if and only if if and only if if and only if if and only if if and only if and if and only if and if and only if and if and only if . The second equality follows similarly.
(11) By (10), . Similarly, .
(12) Since and , we have . Similarly, .
(13) By (pR4) and (pR3), . Hence, . Similarly, we have the second inequality.
(14) Obviously, ; by (pR5), (10), and (4), Hence, .

Next, we continue to investigate the properties of pseudo-weak- algebras and pseudo- algebras, which are needed in the sequel.

Proposition 11. In a pseudo-weak- algebra, the following properties hold:(15), ,(16), ,(17), ,(18),(19), ,(20), ,(21), (22),(23),(24), (25), ,(26), ,(27) if and only if ,(28), ,(29).

Proof. (15) Since , . The second inequality follows similarly.
(16) Since , . The second inequality follows similarly.
(17) Consider . The second inequality follows similarly.
(18) Consider . The second equality follows similarly.
(19) Consider and .
(20) Since and , then . Similarly, we have .
(21) if and only if if and only if . The rest follow similarly.
(22) Consider . The rest follow similarly.
(23) Consider and .
(24) Since and , then , and so . Similarly, . Hence, . On the other hand, by (9), (pR5), (20), and (18), The second equality has a similar proof.
(25) Consider , .
(26) Consider  , .
(27) if and only if if and only if if and only if .
(28) Consider, .
(29) For each , if and only if if and only if if and only if if and only if if and only if if and only if if and only if if and only if if and only if if and only if if and only if .

In a pseudo-weak- algebra (pseudo- algebra) , we define a binary operation as follows, for any :

(30) .

Proposition 12. In a pseudo-weak- algebra, the following properties hold:(31), ,(32),(33),(34) if and only if if and only if ,(35), ,(36), ,(37) if , then and ,(38), ,(39),(40), , whenever the arbitrary unions exist,(41), ,(42), , whenever the arbitrary meets exist,(43), , whenever the arbitrary unions and meets exist,(44),(45),(46),(47), ,(48), .

Proof. (31) Straightforward.
(32) By (pR1) and (pR4), = and .
(33) Consider, .
(34) Consider . And
Consider .
(35) Since , by (34), . Since , by (34), . The second inequality follows similarly.
(36) Since , we have . Since , we have . The second inequality follows similarly.
(37) ; that is, . Similarly, we have that implies that .
(38) By (36), . By (35), . So, . The second inequality follows similarly.
(39) Consider .
(40) For any , for each for each for each . The second equality follows similarly.
(41) Consider . The second equality follows similarly.
(42) For any , for each for each . The second equality follows similarly.
(43) Similarly.
(44) Consider , .
(45) By (41), and . Hence, .
(46) By (40), .
(47) By (13), . Similarly, we have the other one.
(48) By (35), . By (37), ; that is, , and so . The second inequality follows similarly.

Proposition 13. In a pseudo- algebra, the following property holds:
(49)  .

Proof. By (pR1), (5), and (pR6),

Theorem 14. Let A be a pseudo- algebra. If A is a chain, then the operations and can be written by the following:

Proof. If , by (4), . Now, we show that if , then and . By (12), we have and . On the other hand, by (pR6), we have Since , and . By being a chain, we have and , and so and .

4. Categorical Equivalents to Pseudo-IMTL Algebras and Pseudo-NM Algebras

We show that pseudo-weak- algebras coincide with pseudo-IMTL algebras and that pseudo- algebras coincide with pseudo-NM algebras. Hence, as the corollaries of commutative cases, weak- algebras coincide with IMTL-algebras and -algebras coincide with NM-algebras, which are obtained in [18].

Theorem 15. (i) Let be a pseudo-weak- algebra. Define by where . Then, is a pseudo-IMTL algebra.
(ii) Conversely, let be a pseudo-IMTL algebra. Define by Then, is a pseudo-weak- algebra.
(iii) The above defined maps and are mutually inverse.

Proof. (i) (pB1) is verified by (pL1).
(pB2) is verified by (32) and (33).
(pB3) is verified by (34).
(pB4) is verified by (18).
(pB5) is verified by (pL3).
Thus, is a pseudo-IMTL algebra by Definition 3.
(ii) For convenience, in the following proof, we omit the indices “” and “” of the operations.
(pL1): By (pB1).
(pL2): If , by (pB3), and , and so
If , by and (pB3), and , and so
Thus, if , by , and ; that is, and , which means that (pL2) holds.
(pL3): By (pB5).
(pR1): By (pB2) and (pB3), and , and so
Thus, by , and , which means that and .
(pR2): By (pB2), and , and, by (pB3), and . On the other hand, and . Hence, and .
(pR3): By (pB2), (pB3), and , ; that is, , so ; hence, . Similarly, ; that is, , so ; hence, .
(pR4): By (pB2), (pB3), and , , and so ; it follows that . Similarly, we have . Hence, .
(pR5): If , by (pB3), and , and so
Since and , by (pB3), and , and so . Hence, by and (pR4), , so . Similarly, . Namely,
By (pB3) and the same proof of (40), we have whenever the arbitrary unions exist.
By (pB2), (pB4), , and , . On the other hand, by , and , and so . Hence, . Similarly, we have . Namely, (pR5) holds.
Thus, is a pseudo-weak- algebra by Definition 8.
(iii) We put the index “” to the operations of the structure obtained by composition of and .
Let be a pseudo-weak- algebra. We prove that . Indeed,, , , , , and .
Let be a pseudo-IMTL algebra. We prove that . Indeed, , , , and.

Theorem 16. (i) Let be a pseudo- algebra. Define by where . Then, is a pseudo-NM algebra.
(ii) Conversely, let be a pseudo-NM algebra. Define by Then, is a pseudo- algebra.
(iii) The above defined maps and are mutually inverse.

Proof. (i) By Theorem 15 (i), (pB1)–(pB5) hold in . (pB6) is verified by (49).
(ii) By Theorem 15 (ii), (pL1)–(pL3) and (pR1)–(pR5) hold in . Now, we prove that (pR6) also holds in . For convenience, in the following proof, we omit the indices “" and “” of the operations.
By (pB2) and (pB3), for any , and . Hence,
By and (pR4), . Similarly, . Hence,
By (pL2), (pL3), and the same proof of (5) and (6), we have
Thus, by , , and (pB6), we have
(iii) By Theorem 15 (iii).

In Theorems 15 and 16, if , , and , then we have the main results in [18].

Corollary 17 (see [18, Theorem 6]). Weak- algebras coincide with IMTL-algebras.

Corollary 18 (see [18, Theorem 5]). -algebras coincide with NM-algebras.

5. Conclusions

We gave a positive answer to Iorgulescu’s open problem. We extended weak- algebras and -algebras to the noncommutative forms, called pseudo-weak- algebras and pseudo- algebras. The properties of pseudo-weak- algebras and pseudo- algebras were investigated, and the simplified axiom systems of pseudo-weak- algebras and pseudo- algebras were discussed. Finally, we showed that pseudo-weak- algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo- algebras are categorically isomorphic to pseudo-NM algebras. Based on these results, we will study filter theory of pseudo-weak- algebras and pseudo- algebras and 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 authors declare 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).

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