Hall-type representations for generalised orthogroups

An orthogroup is a completely regular orthodox semigroup. The main purpose of this paper is to find a representation of a (generalised) orthogroup with band of idempotents B in terms of a fundamental (generalised) orthogroup. The latter is a subsemigroup of the Hall semigroup W-B (or of its generalisations V-B, U-B and S-B). We proceed in the regular case by constructing a fundamental completely regular subsemigroup (W-B) over bar of W-B, using two different methods. Our subsemigroup plays the role for orthogroups that W-B plays for orthodox semigroups, in that it contains a representation of every orthogroup with band of idempotents B, with kernel of the representation being mu, the greatest congruence contained in H. To develop an analogous theory for classes of generalised orthogroups, that is, to extend beyond the regular case, we replace H by (H) over tilde (B). Generalised orthogroups are then classes of weakly B-superabundant semigroups with (C). We first consider those satisfying an idempotent connected condition (IC) or (WIC). We construct fundamental weakly B-superabundant subsemigroups (V-B) over bar (respectively, (U-B) over bar) of V-B (respectively, U-B) with (C) and (IC) (respectively, with (C) and (WIC)) such that any weakly B-superabundant semigroup with (C) and (IC) (respectively, with (C) and (WIC)) admits a representation to (V-B) over bar (respectively, (U-B) over bar), with kernel of the respresentation being mu(B), the greatest congruence contained in (H) over tilde (B). Finally, we remove the idempotent connected condition and find a representation for an arbitrary weakly B-superabundant semigroup with (C), making use of fresh technology, constructing a fundamental weakly B-superabundant subsemigroup (S-B) over bar of S-B, with the appropriate universal properties. We note that our results are needed in a parallel paper to complete the representation of arbitrary weakly B-superabundant semigroups with (C) as spined products of superabundant Ehresmann semigroups and subsemigroups of S-B.