A Munn type representation of abundant semigroups with a multiplicative ample transversal

The celebrated construction by Munn of a fundamental inverse semigroup $$T_E$$TE from a semilattice E provides an important tool in the study of inverse semigroups and ample semigroups. Munn’s semigroup $$T_E$$TE 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 $$T_E$$TE. 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 $$T_{(I,\Lambda , E^{\circ }, P)}$$T(I,Λ,E∘,P) from a so-called admissible quadruple$$(I,\Lambda , E^{\circ }, P)$$(I,Λ,E∘,P) that plays for abundant semigroups with a multiplicative ample transversal the role that $$T_E$$TE 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 $$(I,\Lambda , E^{\circ }, P)$$(I,Λ,E∘,P) if and only if it is embeddable as a full subsemigroup (resp. full regular subsemigroup) into $$T_{(I,\Lambda , E^{\circ }, P)}$$T(I,Λ,E∘,P). Our results generalize and enrich some classical results of Munn on inverse semigroups and of Fountain on ample semigroups.