Document Type

Article

Language

eng

Format of Original

11 p.

Publication Date

10-2014

Publisher

Springer

Source Publication

Semigroup Forum

Source ISSN

0037-1912

Original Item ID

doi: 10.1007/s00233-014-9575-2

Abstract

Yu, Wang, Wu and Ye call a semigroup S τ -congruence-free, where τ is an equivalence relation on S, if any congruence ρ on S is either disjoint from τ or contains τ . A congruence-free semigroup is then just an ω-congruence-free semigroup, where ω is the universal relation. They determined the completely regular semigroups that are τ -congruence-free with respect to each of the Green’s relations. The goal of this paper is to extend their results to all regular semigroups. Such a semigroup is J –congruence-free if and only if it is either a semilattice or has a single nontrivial J -class, J, say, and either J is a subsemigroup, in which case it is congruence-free, or otherwise its principal factor is congruence-free. Given the current knowledge of congruence-free regular semigroups, this result is probably best possible. When specialized to completely semisimple semigroups, however, a complete answer is obtained, one that specializes to that of Yu et al. A similar outcome is obtained for L and R. In the case of H, only the completely semisimple case is fully resolved, again specializing to those of Yu et al.

Comments

Accepted version. Semigroup Forum, Vol. 89, No. 2 (October 2014): 383-393. © 2014 Springer. Used with permission.

The final publication is available at Springer via DOI.

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