Letting κ be the minimum
unitary-congruence on a regular semigroup S and ξ be the minimum congruence such
that S∕ξ is a semilattice of groups, it is the purpose of this paper to characterize all
regular semigroups for which κ ∩ ξ is the identity relation. That is, we describe all
regular semigroups which are subdirect products of a unitary semigroup and a
semilattice of groups. In the process of doing this, a description of ξ is given for any
orthodox semigroup.
In A. H. Clifford’s paper on radicals in semigroups, [2], a diagram was given
presenting the relationship between various classes of regular semigroups and certain
minimum congruences. Two questions were left open. The first was to find all
subdirect products of a band and a semilattice of groups, that is, all semigroups for
which β ∩ ξ is the identity, where β is the minimum band-congruence. This was
solved by Schein in [14] and also by Petrich in Theoremi 3.2 of [11]. The second
question involves finding all subdirect products of a semilattice of groups and a
regular semigroup whose set of idempotents is unitary. In this paper we
find that any such semigroup can be described as a semilattice of unitary
semigroups on which ℋ∩ σ is a unitary-congruence, where σ is the minimum
group-congruence. A description will also be given in terms of restrictions on the
structure homomorphisms. In order to accomplish this, we first give an explicit
characterization of ξ, the minimum semilattice of groups-congruence, on any
orthodox semigroup.
