Regular Semigroups with Quasi-Ideal Orthodox Transversals

A New Construction for Regular Semigroups With Quasi-Ideal Orthodox Transversals

“…In what follows R denotes a regular semigroup with a right ideal orthodox transversal S o . Then by [7,Lemma 1], E(R) = I is a band, consequently R is an orthodox semigroup and we will denote the minimum inverse semigroup congruence on R by γ . For a ∈ R, the R-class of R containing a will be denoted by R a and the γ -class containing a will be denoted by T (a).…”

“…In a previous publication [7] we constructed regular semigroups with quasi-ideal orthodox transversals by a formal set (B, R), where R is a regular semigroup with a right ideal orthodox transversal S o and B a band with a left ideal orthodox (in fact, band) transversal E o . Evidently, there are different conditions on the structural 'brick' B and R. The present paper corrects this asymmetry by giving a new construction of regular semigroups with quasi-ideal orthodox transversals by way of two regular semigroups R and L. The semigroups R and L share a common orthodox transversal S o , which is a right ideal of R and a left ideal of L. Many of the conditions on R and L are symmetric and one is weaker than that in [7] (that is, if x ∈ S o or a ∈ S o then a * x = ax in this paper; instead of if x ∈ E o or e ∈ E o , then e * x = ex in [7]).…”

On Generalized Orthodox Transversals

“…Afterwards, Chen and Guo [5] considered the general case of orthodox transversals and investigated some properties concerned with the sets I and . In [6,7], two new structure theorems for regular semigroups with quasi-ideal orthodox transversals were also established and in [8], left simplistic orthodox transversals were studied. In 2008, Zhang and Wang [9] introduced and studied generalized inverse 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.…”