On Cyclic Associative Semihypergroups and Neutrosophic Extended Triplet Cyclic Associative Semihypergroups

2022

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In this paper, we study the connection between generalized quasi-left alter BCI-algebra and commutative Clifford semigroup by introducing the concept of an adjoint semigroup. We introduce QM-BCI algebra, in which every element is a quasi-minimal element, and prove that each QM-BCI algebra is equivalent to generalized quasi-left alter BCI-algebra. Then, we introduce the notion of generalized quasi-left alter-hyper BCI-algebra and prove that every generalized quasi-left alter-hyper BCI-algebra is a generalized quasi-left alter BCI-algebra. Next, we propose a new notion of quasi-hyper BCI algebra and discuss the relationship among them. Moreover, we study the subalgebras of quasi-hyper BCI algebra and the relationships between Hv-group and quasi-hyper BCI-algebra, hypergroup and quasi-hyper BCI-algebra. Finally, we propose the concept of a generalized quasi-left alter quasi-hyper BCI algebra and QM-quasi hyper BCI-algebra and discuss the relationships between them and related BCI-algebra.

Section: Discussionmentioning

confidence: 99%

A Class of BCI-Algebra and Quasi-Hyper BCI-Algebra

2022

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“…This helps us to prove that an algebraic structure is a completely regular semigroup, which requires fewer steps and is more convenient. As the next research topic, we can explore the relationships among transposition regular semigroups and hypersemigroups and non-classical logical algebras (see [31][32][33]).…”

Section: Discussionmentioning

confidence: 99%

Left (Right) Regular and Transposition Regular Semigroups and Their Structures

2022

Mathematics

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Regular semigroups and their structures are the most wonderful part of semigroup theory, and the contents are very rich. In order to explore more regular semigroups, this paper extends the relevant classical conclusions from a new perspective: by transforming the positions of the elements in the regularity conditions, some new regularity conditions (collectively referred to as transposition regularity) are obtained, and the concepts of various transposition regular semigroups are introduced (L1/L2/L3, R1/R2/R3-transposition regular semigroups, etc.). Their relations with completely regular semigroups and left (right) regular semigroups, proposed by Clifford and Preston, are analyzed. Their properties and structures are studied from the aspects of idempotents, local identity elements, local inverse elements, subsemigroups and so on. Their decomposition theorems are proved respectively, and some new necessary and sufficient conditions for semigroups to become completely regular semigroups are obtained.

“…In this paper, we mainly talk about the various transposition regular AG-groupoids. We can discuss the relationships among transposition regular AG-groupoids, hypersemigroups and T2CA-groupoids as well as non-classical logical algebras (see [24][25][26][27][28]).…”

Section: Discussionmentioning

confidence: 99%

Transposition Regular AG-Groupoids and Their Decomposition Theorems

2022

Mathematics

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In this paper, we introduce transposition regularity into AG-groupoids, and a variety of transposition regular AG-groupoids (L1/R1/LR, L2/R2/L3/R3-groupoids) are obtained. Their properties and structures are discussed by their decomposition theorems: (1) L1/R1-transposition regular AG-groupoids are equivalent to each other, and they can be decomposed into the union of disjoint Abelian subgroups; (2) L1/R1-transposition regular AG-groupoids are LR-transposition regular AG-groupoids, and an example is given to illustrate that not every LR-transposition regular AG-groupoid is an L1/R1-transposition regular AG-groupoid; (3) an AG-groupoid is an L1/R1-transposition regular AG-groupoid if it is an LR-transposition regular AG-groupoid satisfying a certain condition; (4) strong L2/R3-transposition regular AG-groupoids are equivalent to each other, and they are union of disjoint Abelian subgroups; (5) strong L3/R2-transposition regular AG-groupoids are equivalent to each other and they can be decomposed into union of disjoint AG subgroups. Their relations are discussed. Finally, we introduce various transposition regular AG-groupoid semigroups and discuss the relationships among them and the commutative Clifford semigroup as well as the Abelian group.