Vol. 64, No. 1, 1976

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Certain congruences on orthodox semigroups

Janet E. Mills

Vol. 64 (1976), No. 1, 217–226

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.

Mathematical Subject Classification 2000
Primary: 20M10
Received: 29 September 1975
Published: 1 May 1976
Janet E. Mills
Department of Mathematics
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