Congruence semimodular varieties II: Regular varieties

Abstract

In [8], P. Jones characterized the regular varieties of semigroups which are congruence semimodular and he partially solved the problem of characterizing the non-regular, congruence semimodular (CSM) varieties of semigroups. For regular varieties his characterization was an equational one, so necessarily a part of his argument was semigroup-theoretic. But some of his argument involved only congruence lattice manipulations and references to the two-element semilattice. It seemed plausible that one could give a universal algebraic characterization of all regular, CSM varieties. Jones posed the problem of characterizing regular, CSM varieties at the International Conference on Universal Algebra and Lattice Theory held at Molokai, Hawaii in 1987. The authors were students attending that conference and became familiar with the problem. Agliano began a general investigation of CSM varieties under the supervision of his doctoral advisor, J. B. Nation, at the University of Hawaii. In the fall of 1988, Agtiano filed his dissertation and Kearnes arrived at the University of Hawaii. We began discussing whether or not tame congruence theory could be applied to solve Jones' problem, at least for locally finite varieties or pseudo-varieties of finite algebras. This is the approach that had been suggested by Jones in 1987. We discovered a number of interesting facts about CSM varieties using tame congruence theory, but we did not solve Jones' problem at that time. (These "interesting facts" have since been collected in, Congruence semimodular varieties I: locally finite varieties.) After about two months our collaboration ended when Agliano returned to Italy. For a while we felt compelled to publish some of our results on CSM varieties, but we were reluctant to do so without solving Jones' problem first. We renewed