A completely inverse AG

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 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 E-unitary semigroups. In particular, we prove that in any E-inversive semigroup, H ∩ σ ⊆ κ, 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, H ∩ σ = 1 S .

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We study some special types of bands of E-inversive unipotent semigroups. It has been proved that in any R-semigroup S, which is a band of E-inversive unipotent semigroups, the set of its regular elements is a retract of S. Also, some characterizations of E-inversive rectangular bands of unipotent semigroups are given. This theorem extends nearly 40-old results from the theory of epigroups. In fact, a more general result is valid in some special subclass of the class of E-inversive semigroups; this result seems to be (partially) new for all epigroups.