Hall-type representations for generalised orthogroups

Abstract

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 $\overline{W_{B}}$ 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 μ, the greatest congruence contained in $\mathcal{H}$.

To develop an analogous theory for classes of generalised orthogroups, that is, to extend beyond the regular case, we replace $\mathcal{H}$ by $\widetilde{\mathcal{H}}_{B}$. 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 $\overline{V_{B}}$ (respectively, $\overline{U_{B}}$) 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 $\overline{V_{B}}$ (respectively, $\overline{U_{B}}$), with kernel of the respresentation being μ _B , the greatest congruence contained in $\widetilde{\mathcal{H}}_{B}$. 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 $\overline{S_{B}}$ 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 .