Fundamental Representations for Classes of Semigroups Containing a Band of Idempotents

Abstract

The construction by Hall of a fundamental orthodox semigroup W B from a band B provides an important tool in the study of orthodox semigroups. Hall's semigroup W B has the property that a semigroup is fundamental and orthodox with band of idempotents isomorphic to B if and only if it is embeddable as a full subsemigroup into W B . The aim of this article is to extend Hall's approach to some classes of nonregular semigroups. From a band B, we construct a semigroup U B that plays the role of W B for a class of weakly B-abundant semigroups having a band of idempotents B. The semigroups we consider, in particular U B , must also satisfy a weak idempotent connected condition. We show that U B has subsemigroup V B where V B satisfies a stronger notion of idempotent connectedness, and is again the canonical semigroup of its kind. In turn, V B contains W B as its subsemigroup of regular elements. Thus we have the following inclusions as subsemigroups: either of which may be strict, even in the finite case. The existence of the semigroups U B and V B enable us to prove a structure theorem for classes of weakly B-abundant semigroups having band of idempotents B, and satisfying either of our idempotent connected conditions, as spined products of U B , or V B , with a weakly B/ 𝒟 -ample semigroup.