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

Abstract

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.

Transposition Regular AG-Groupoids and Their Decomposition Theorems

“…In [19], for a semigroup S, for any element a ∈ S, there exists p, q ∈ S such that (pa)a = a = a(aq). However, in an AG-groupoid, we can get the whole equation through the right equation.…”

“…In [19], they introduced various transposition regular semigroups and prove that L1/R1/LR-transposition regular semigroups and a completely regular semigroup are equivalent to each other. Definition 18.…”

“…An AG-groupoid is the generalization of a semigroup theory and has vast applications in collaboration with semigroups (see [18]). Based on the close connections between semigroups and AGgroupoids, we introduce the transposition regularity proposed by Xiaohong Zhang and Yudan Du in [19] into AG-groupoids.…”

Transposition Regular TA-Groupoids and Their Structures

“…Cattaneo and Contreras defined a regular relational symplectic groupoid and showed that every Poisson manifold arises as the "space of objects" of a regular relational symplectic groupoid in [16]. In [17], Xiaohong Zhang et al proposed a new research method to study semigroups, that is, introducing the concepts of various transposition regular semigroups and studying their structures. The successful application of this new transposition regular research method in the Abel-Grassmann's groupoid (AG-groupoid) [18] also prompted us to apply it to the TA-groupoid.…”

“…The successful application of this new transposition regular research method in the Abel-Grassmann's groupoid (AG-groupoid) [18] also prompted us to apply it to the TA-groupoid. As a continuation of [17,18], we propose the notions of transposition regular TA-groupoids and investigate their properties and structural characteristics. This is also the embodiment of the transposition regularity method in the TA-groupoid.…”