A new construction for abundant semigroups with multiplicative quasi-adequate transversals

Abstract

In any abundant semigroup with a quasi-adequate transversal, we define two sets R and L and give some properties and characterizations associated with them. Then we give a structure theorem for abundant semigroups with multiplicative quasi-adequate transversals by means of two quasi-adequate semigroups R and L.

On refined generalised quasi-adequate transversals

“…The concept of quasi-adequate transversals was introduced by Ni [19] and followed by Luo, Kong and Wang [20,21], their work mainly focused on the properties and the structure of multiplicative quasiadequate transversals. Unfortunately, quasi-adequate transversals are neither the generalisations of orthodox transversals nor the generalisations of adequate 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.…”

“…The concept of adequate transversals was introduced for abundant semigroups by El-Qallali [5] as an analogue of inverse transversals, and followed by Chen, Guo, Shum, Kong and Wang etc. [3,11,14,18,19]. In [19], the authors shown that the product of any two quasi-ideal adequate transversals of an abundant semigroup S which satisfy the regularity condition is a quasi-ideal adequate transversal of S.…”