The celebrated construction by Munn of a fundamental inverse semigroup \(T_E\) from a semilattice E provides an important tool in the study of inverse semigroups and ample semigroups. Munn’s semigroup \(T_E\) has the property that a semigroup is a fundamental inverse semigroup (resp. a fundamental ample semigroup) with a semilattice of idempotents isomorphic to E if and only if it is embeddable as a full inverse subsemigroup (resp. a full subsemigroup) into \(T_E\). The aim of this paper is to extend Munn’s approach to a class of abundant semigroups, namely abundant semigroups with a multiplicative ample transversal. We present here a semigroup \(T_{(I,\Lambda , E^{\circ }, P)}\) from a so-called admissible quadruple\((I,\Lambda , E^{\circ }, P)\) that plays for abundant semigroups with a multiplicative ample transversal the role that \(T_E\) plays for inverse semigroups and ample semigroups. More precisely, we show that a semigroup is a fundamental abundant semigroup (resp. fundamental regular semigroup) having a multiplicative ample transversal (resp. multiplicative inverse transversal) whose admissible quadruple is isomorphic to \((I,\Lambda , E^{\circ }, P)\) if and only if it is embeddable as a full subsemigroup (resp. full regular subsemigroup) into \(T_{(I,\Lambda , E^{\circ }, P)}\). Our results generalize and enrich some classical results of Munn on inverse semigroups and of Fountain on ample semigroups.
['Shoufeng Wang']
A Munn type representation of abundant semigroups with a multiplicative ample transversal