On Adequate Transversals

Abstract

We consider adequate transversals of abundant semigroups and prove that, in a particular case, there is a natural embedding of an inverse transversal within a certain regular subsemigroup. We also introduce the concepts of simplistic, perfect and quasi-adequate transversal and provide a number of interesting connections between these.

Semi-abundant semigroups with quasi-Ehresmann transversals

“…(3) Letx ∈ Γ x and e • ∈ Γ x ∩ E(S • ). Then, e • δx by (1). Hence, there exist k, l ∈ E(e • ) such thatx = ke • l, which implies thatx ∈ E(S • ).…”

“…The concept of inverse transversals was introduced by Blyth-McFadden [3]. From then on, inverse transversals have been extensively investigated and generalized by many authors (for example, see [1]- [7], [14]- [15] and [18]). Since orthodox semigroups can be regarded as generalizations of inverse semigroups, in 1999, Chen [4] generalized inverse transversals to orthodox transversals in the class of regular semigroups and gave a construction theorem for a kind of regular semigroups with orthodox transversals.…”

On Naturally Ordered Abundant Semigroups with an Adequate Monoid Transversal

“…Let ∘ be a * -adequate subsemigroup of and ∈ ∘ . Throughout this paper, we denote by + [resp., Lemma 5 (see [6]). Let ∘ be an adequate transversal of an abundant semigroup and , ∈ .…”

“…In what follows, we shall use the notion and notation of [6,21]. Other undefined terms can be found in [11,22,23].…”

“…Since abundant semigroups generalize regular semigroups, it is natural that, in the first instance, we should approach their structure theory by looking for generalization of results from the theory of regular semigroups. The following papers contain some work along these lines: [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. As we shall discuss below, this paper is one of a sequence in which we concentrate on the structure of a class of naturally ordered abundant semigroups with an adequate monoid transversal.…”