On Generalized Orthodox Transversals

Abstract

Generalized orthodox transversals are introduced in this article, and some characterizations associated with them are obtained. The related results of Chen on quasi-ideal orthodox transversals obtained in 1999 and Zhang and Wang on generalized inverse transversals obtained in 2008 are generalized and amplified. A structure theorem of a regular semigroup with a perfect generalized orthodox transversal is also established.

On refined generalised quasi-adequate transversals

“…since aλ a x o ∈ S o ΛS o ⊆ S o . Hence ax o ∈ R by Proposition 2.5 and SS o ⊆ R. Dually S o S ⊆ L.(8) =⇒(9). For any a ∈ S, x ∈ R, x = i x x o λ x with λ x ∈ E o , we have ax = ai x • x o λ x ∈ SS o ⊆ R and R is a left ideal of S. Dually, S o S ⊆ L implies that L is a right ideal of S.(9) =⇒ (1).…”

“…In [7,8], Kong and Zhao introduced two interesting sets R and L and established the structure theorems for regular semigroups with quasi-ideal orthodox transversals. In 2014, Kong [9] introduced the concept of generalised orthodox transversals and obtained some basic properties associated with them. Kong and Meng [10] acquired the characterization for generalised orthodox transversals to be orthodox transversals.…”

The characterization and the product of quasi-Ehresmann transversals

“…The concept of inverse transversals of regular semigroups was introduced by Blyth-McFadden [1]. Since then, inverse transversals have attracted much attention and a series of important results have been obtained and generalized (see [1][2][3][4][5]11,[13][14][15][16][17][18][19][20][21][23][24][25][26]). If S is a regular semigroup, then an inverse transversal of S is an inverse subsemigroup S o which meets V(a) precisely once for each a ∈ S (that is, |V(a) ∩ S o | = 1), where V(a) = {x ∈ S| axa = a and xax = x} denotes the set of inverses of a.…”

“…Chen-Guo [4] obtained some important properties associated with orthodox transversals in the general case. Most recently, Kong, Meng, Zhao [13,15,16,17,21] investigated orthodox transversals and obtained some interesting results. Especially, Kong-Meng [17] acquired the characterization for a generalized orthodox transversal to be an orthodox transversal and present a concrete description of the maximum idempotent separating congruence on regular semigroups with orthodox transversals.…”