Congruences and group congruences on a semigroup.
Let \(S\) be a semigroup and \(E\) be the set of idempotents of \(S\). A subsemigroup \(N\) of \(S\) is called normal if it is full (i.e., \(E\subseteq S\)), dense (i.e., \((\forall s\in S)(\exists x,y\in S)(sx,ys\in N)\)), reflexive (i.e., \((\forall a,b\in S)(ab\in N\Rightarrow ba\in N)\)) and closed in \(S\) (i.e., \(N\omega=N\), where \(N\omega=\{s\in S:(\exists a\in N)(as\in N)\)). A semigroup \(S\) is said to be \(E\)-inversive if for every \(a\in S\) there exists \(x\in S\) such that \(ax\in E\). \(S\) is an \(E\)-semigroup if \(E\) is a subsemigroup of \(S\). Regular \(E\)-semigroups are called orthodox. A semigroup with \(E\neq\emptyset\) is said to be \(E\)-unitary if \(E\) is unitary subset of \(S\), i.e. \(se\in E\) implies \(s\in E\) and \(es\in E\) implies \(s\in E\) for every \(s\in S\), \(e\in E\). The author shows 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 \(S\). Also group congruences on \(E\)-inversive (\(E\)-)semigroups are described. In particular, these results generalize the result of \textit{J. Meakin} [J. Aust. Math. Soc. 13, 259-266 (1972; Zbl 0265.20056)] concerning the description of the least group congruence on an orthodox semigroup, the result of \textit{J. M. Howie} [Proc. Edinb. Math. Soc., II. Ser. 14, 71-79 (1964; Zbl 0123.01701)] concerning the description of \(\rho\vee\sigma\) in an inverse semigroup \(S\), where \(\rho\) is a congruence and \(\sigma\) is the least group congruence on \(S\), and some other known results. Also, one of the main aims of the reviewed paper is to study group congruences on \(E\)-unitary semigroups. In particular, it is proved that in any \(E\)-inversive semigroup, \(\mathcal H\cap\sigma\subseteq\kappa\), where \(\kappa\) is the least \(E\)-unitary congruence. This result is equivalent to the statement that in an arbitrary \(E\)-unitary \(E\)-inversive semigroup \(S\), \(\mathcal H\cap\sigma=1_S\).
scientific article; zbMATH DE number 6176653
scientific article