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:
W-B subset of V-B subset of U-B,
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.
- hall semigroup
- idempotent connected
- weakly abundant
- ORTHODOX SEMIGROUPS