Pseudo-Weak-R 0 Algebras

A positive answer to the open problem of Iorgulescu on extending weak-R 0 algebras and R 0-algebras to the noncommutative forms is given. We show that pseudo-weak-R 0 algebras are categorically isomorphic to pseudo-IMTL algebras and that pseudo-R 0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-R 0 algebras and R 0-algebras, are investigated. The simplified axiom systems of pseudo-weak-R 0 algebras and pseudo-R 0 algebras are obtained.


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 pseudologic 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 2 The Scientific World Journal the structure of the real unit interval [0, 1], induced by a left-continuous t-norm. Flondor et al. [13] presented pseudo-MTL-algebras as a noncommutative generalization of MTLalgebras.
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. 0 -algebras were introduced by Wang [14] as the algebraic structure for his formal deductive system * of fuzzy propositional calculus. Weak-0 -algebras [14] are the generalization of 0algebras. In the recent years, the research on 0 -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-0 algebras, introduced in 1997 by Wang, and that NM-algebras are categorically isomorphic to 0 -algebras, introduced also in 1997 by Wang. Extend weak-0 algebras and 0 -algebras to the noncommutative case.
In this paper, we extend weak-0 algebras and 0algebras to the noncommutative forms, called pseudo-weak-0 algebras and pseudo-0 algebras. We show that pseudoweak-0 algebras are categorically isomorphic to pseudo-IMTL-algebras and that pseudo-0 algebras are categorically isomorphic to pseudo-NM algebras. Some properties, the noncommutative forms of the properties in weak-0 algebras and 0 -algebras, are investigated. Furthermore, we discuss the simplified axiom systems of pseudo-weak-0 algebras and pseudo-0 algebras.

Preliminaries
We recall some definitions and results which will be used in the sequel.
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 orderreversing involution with respect to ≤, and the following conditions hold for any , , ∈ : where 1 is the largest element of , then one calls a weak-0 algebra. An 0 algebra is a weak-0 algebra satisfying the following additional condition:

Pseudo-Weak-0 Algebras and Pseudo-0 Algebras
We introduce the notions of pseudo-weak-0 algebras and pseudo-0 algebras. They are noncommutative forms of weak-0 algebras and 0 -algebras. Some of their properties are investigated.
(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 ( , ∧, ∨, 0, 1), Hence, we have the following simplified definition. In what follows, we will use the following definition.
Next, we continue to investigate the properties of pseudoweak-0 algebras and pseudo-0 algebras, which are needed in the sequel.

Proposition 11.
In a pseudo-weak-0 algebra, the following properties hold: The second equality has a similar proof. In a pseudo-weak-0 algebra (pseudo-0 algebra) , we define a binary operation ⊙ as follows, for any , ∈ :

Categorical Equivalents to Pseudo-IMTL Algebras and Pseudo-NM Algebras
We show that pseudo-weak-0 algebras coincide with pseudo-IMTL algebras and that pseudo-0 algebras coincide with pseudo-NM algebras. Hence, as the corollaries of commutative cases, weak-0 algebras coincide with IMTLalgebras and 0 -algebras coincide with NM-algebras, which are obtained in [18].

Conclusions
We gave a positive answer to Iorgulescu's open problem. We extended weak-0 algebras and 0 -algebras to the noncommutative forms, called pseudo-weak-0 algebras and pseudo-