Fundamental Semigroups Having a Band of Idempotents

The construction by Hall of a fundamental orthodox semigroup WB from a band B provides an important tool in the study of orthodox semigroups. We present here a semigroup SB that plays the role of WB for a class of semigroups having a band of idempotents B. Specifically, the semigroups we consider are weakly B-abundant and satisfy the congruence condition (C). Any orthodox semigroup S with E(S) = B lies in our class. On the other hand, if a semigroup S lies in our class, then S is Ehresmann if and only if B is a semilattice. The Hall semigroup WB is a subsemigroup of SB , as are the (weakly) idempotent connected semigroups VB and UB. We show how the structure of SB can be used to extract information relating to arbitrary weakly B-abundant semigroups with (C).