Fundamental representations for classes of semigroups containing a band of idempotents

TY - JOUR

T1 - Fundamental representations for classes of semigroups containing a band of idempotents

AU - El Qallali, Abdulsalam

AU - Fountain, John

AU - Gould, Victoria

PY - 2008/8/8

Y1 - 2008/8/8

N2 - 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.

AB - 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.

KW - band

KW - fundamental

KW - hall semigroup

KW - idempotent connected

KW - weakly abundant

KW - ORTHODOX SEMIGROUPS

UR - http://www.scopus.com/inward/record.url?scp=50149111115&partnerID=8YFLogxK

U2 - 10.1080/00927870802110649

DO - 10.1080/00927870802110649

M3 - Article

VL - 36

SP - 2998

EP - 3031

JO - Communications in Algebra

JF - Communications in Algebra

IS - 8

ER -