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The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal

1
Department of Mathematics, Shaanxi University of Science & Technology, Xi’an 710021, China
2
Department of Mathematics, Shanghai Maritime University, Shanghai 201306, China
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(10), 465; https://doi.org/10.3390/sym10100465
Received: 19 September 2018 / Revised: 28 September 2018 / Accepted: 29 September 2018 / Published: 8 October 2018
(This article belongs to the Special Issue Discrete Mathematics and Symmetry)
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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. View Full-Text
Keywords: fuzzy logic; pseudo-BCI algebra; quasi-maximal element; KG-union; quasi-alternating BCK-algebra fuzzy logic; pseudo-BCI algebra; quasi-maximal element; KG-union; quasi-alternating BCK-algebra
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Wu, X.; Zhang, X. The Structure Theorems of Pseudo-BCI Algebras in Which Every Element is Quasi-Maximal. Symmetry 2018, 10, 465.

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