Abstract
For mathematical fuzzy logic systems, the study of corresponding algebraic structures plays an important role. Pseudo-BCI algebra is a class of non-classical logic algebras, which is closely related to various non-commutative fuzzy logic systems. The aim of this paper is focus on the structure of a special class of pseudo-BCI algebras in which every element is quasi-maximal (call it QM-pseudo-BCI algebras in this paper). First, the new notions of quasi-maximal element and quasi-left unit element in pseudo-BCK algebras and pseudo-BCI algebras are proposed and some properties are discussed. Second, the following structure theorem of QM-pseudo-BCI algebra is proved: every QM-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an anti-group pseudo-BCI algebra. Third, the new notion of weak associative pseudo-BCI algebra (WA-pseudo-BCI algebra) is introduced and the following result is proved: every WA-pseudo-BCI algebra is a KG-union of a quasi-alternating BCK-algebra and an Abel group.
Highlights
In the study of t-norm based fuzzy logic systems [1,2,3,4,5,6,7,8,9], algebraic systems play an important role.In this paper, we discuss pseudo-BCI/BCK algebras which are connected with non-commutative fuzzy logic systems.BCK-algebras and BCI-algebras were introduced by Iséki [10] as algebras induced by Meredith’s implicational logics BCK and BCI
We discuss pseudo-BCI/BCK algebras which are connected with non-commutative fuzzy logic systems
By using the notions of quasi-maximal element, quasi-left unit element, KG-union and direct product, we give the structure theorem of the class of pseudo-BCI algebras in which every element is quasi-maximal
Summary
In the study of t-norm based fuzzy logic systems [1,2,3,4,5,6,7,8,9], algebraic systems (such as residuated lattices, BL-algebras, MTL-algebras, pseudo-BL algebras, pseudo-MTL algebras, et al.) play an important role. We discuss pseudo-BCI/BCK algebras which are connected with non-commutative fuzzy logic systems (such that non-commutative residuared lattices, pseudo-BL/pseudo-MTL algebras). The notion of pseudo-BCK algebra was introduced by G. We further study the structure characterizations of pseudo-BCI algebras. By using the notions of quasi-maximal element, quasi-left unit element, KG-union and direct product, we give the structure theorem of the class of pseudo-BCI algebras in which every element is quasi-maximal (call they QM-pseudo-BCI algebras). It should be noted that the original definition of pseudo-BCI/BCK algebra is different from the definition used in this paper. We think that the logical semantics of this algebraic structure can be better represented by using the present definition
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