Congruences on Regular Semigroups

Let S be a regular semigroup and let p be a congruence relation on S. The kernel of p, in notation kerp, is the union of the idempotent p-classes. The trace of p, in notation trp, is the restriction of p to the set of idempotents of S. The pair (kerp,trp) is said to be the congruence pair associated with p. Congruence pairs can be characterized abstractly, and it turns out that a congruence is uniquely determined by its associated congruence pair. The triple ((pV £■)/£, kerp, (pV %)/%) is said to be the congruence triple associated with p. Congruence triples can be characterized abstractly and again a congruence relation is uniquely determined by its associated triple. The consideration of the parameters which appear in the above-mentioned representations of congruence relations gives insight into the structure of the congruence lattice of S. For congruence relations p and 8, put pT¡0 [pTr8, pT8] if and only if p V £ = 9 V L [p V R = 8 V £,trp = tröj. Then Th Tr and T are complete congruences on the congruence lattice of S and T = Ti D Tr. Introduction and summary. After it was realized by Wagner that a congruence on an inverse semigroup S is uniquely determined by its idempotent classes, Preston provided an abstract characterization of such a family of subsets of S called the kernel normal system (see [2, Chapter 10]). This approach was the only usable means for handling congruences on inverse semigroups for two decades. A new approach to the problem of describing congruences on inverse semigroups was sparked by the work of Scheiblich [13] who described congruences in terms of kernels and traces. A systematic exposition of the achievements of this approach can be found in [10, Chapter III]. It was recognized by Feigenbaum [3] that every congruence p on a regular semigroup S is uniquely determined by its kernel, kerp, equal to elements p-equivalent to idempotents, and its trace, trp, equal to the restriction of p to the set E(S) of idempotents of S. In the case of an inverse semigroup S, kerp and trp, as well as their mutual relationship, can be described abstractly by means of simple conditions on a subset of S and an equivalence on E(S) (see [10, Chapter III]). Following in the footsteps of Scheiblich, for orthordox and arbitrary regular semigroups, Feigenbaum [3] and Trotter [14] adopted the following approach: trp is characterized abstractly and to each such trp all matching kernels are described. This unbalances the symmetry of the kernel-trace approach by giving preference to the trace. Hence a balanced view relative to the kernel and the trace is evidently called for. The unqualified success of the kernel-trace approach for inverse semigroups, including its diverse ramifications, gave a certain hope that this may also turn out to be the case for regular semigroups. Judging by the complexity of regular semigroups and the attempts made for both orthodox and general regular semigroups, Received by the editors November 2, 1984. 1980 Mathematics Subject Classification. Primary 20M10; Secondary 08A30. ©1986 American Mathematical Society 0002-9947/86 $1.00 + $.25 per page