Congruences and group congruences on a semigroup
 Roman S. Gigoń
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Abstract
We show that there is an inclusionpreserving 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 Einversive (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 ρ∨σ 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 Eunitary semigroups. In particular, we prove that in any Einversive semigroup, \(\mathcal{H}\cap\sigma\subseteq\kappa\) , where κ is the least Eunitary congruence. This result is equivalent to the statement that in an arbitrary Eunitary Einversive semigroup S, \(\mathcal{H}\cap\sigma= 1_{S}\) .
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 Title
 Congruences and group congruences on a semigroup
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Semigroup Forum
Volume 86, Issue 2 , pp 431450
 Cover Date
 20130401
 DOI
 10.1007/s002330129425z
 Print ISSN
 00371912
 Online ISSN
 14322137
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Group congruence
 Einversive semigroup
 Esemigroup
 Idempotentsurjective semigroup
 Eventually regular semigroup
 Idempotent pure congruence
 Idempotentseparating congruence
 Eunitary congruence
 Authors

 Roman S. Gigoń ^{(1)}
 Author Affiliations

 1. Institute of Mathematics and Computer Science, Wroclaw University of Technology, Wyb. Wyspianskiego 27, 50370, Wroclaw, Poland