On regular semigroups with orthodox transversals

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

“…(1) =⇒ (3). If (1) holds, then for any y ∈ S, x o ∈ S o , we have (4) =⇒ (2). If (4) holds, then for any f ∈ Λ and e ∈ I, there exist h, l ∈ E o such that hRf and lLe.…”

Adequate Transversals of Quasi-Adequate Semigroups

“…As one might expect a number of generalisations of this concept have emerged and connections and similarlities with inverse transversals studied. For example, we may replace the inverse subsemigroup with a different type of regular subsemigroup such as an orthodox semigroup and, with a suitable adjustment of the definitions, refer to orthodox transversals of regular semigroups (see for example [6]). Or we may replace the unique inverse x 0 of the regular element x with another related element such as an associate -recall that an associate of an element x is an element y such that xyx = x -and study the structure of these associate inverse subsemigroups (see for example [3]).…”