Semigroup Forum

, Volume 86, Issue 2, pp 431-450

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

Congruences and group congruences on a semigroup

  • Roman S. GigońAffiliated withInstitute of Mathematics and Computer Science, Wroclaw University of Technology Email author 


We show that there is an inclusion-preserving bijection between the set of all normal subsemigroups of a semigroup S and the set of all group congruences on S. We describe also group congruences on E-inversive (E-)semigroups. In particular, we generalize the result of Meakin (J. Aust. Math. Soc. 13:259–266, 1972) concerning the description of the least group congruence on an orthodox semigroup, the result of Howie (Proc. Edinb. Math. Soc. 14:71–79, 1964) concerning the description of ρσ in an inverse semigroup S, where ρ is a congruence and σ is the least group congruence on S, some results of Jones (Semigroup Forum 30:1–16, 1984) and some results contained in the book of Petrich (Inverse Semigroups, 1984). Also, one of the main aims of this paper is to study of group congruences on E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\), where κ is the least E-unitary congruence. This result is equivalent to the statement that in an arbitrary E-unitary E-inversive semigroup S, \(\mathcal{H}\cap\sigma= 1_{S}\).


Group congruence E-inversive semigroup E-semigroup Idempotent-surjective semigroup Eventually regular semigroup Idempotent pure congruence Idempotent-separating congruence E-unitary congruence