Congruences on strong semilattices of regular simple semigroups (Q1101833)

In this thesis [Univ. Wien, 1982; Zbl 0474.20035] \textit{K. Auinger} studied congruences on strong semilattices Y of regular and simple semigroups \(S_{\alpha}\) (\(\alpha\in Y)\). In this paper the author adopts a similar development for the construction of all congruences considering congruences on Y and for each \(\alpha\in Y\) a congruence on \(S_{\alpha}\) satisfying certain simple compatibility conditions. (This development does not make use of the kernel-trace approach introduced by \textit{F. Pastijn} and \textit{M. Petrich} [in Trans. Am. Math. Soc. 295, 607- 633 (1986; Zbl 0599.20095)].) First, a characterization of such semigroups in terms of the natural partial order on a regular semigroup is given. Using the main characterization theorem for congruences on strong semilattices of regular and simple semigroups, S, the special case is studied, where all the linking homomorphisms different from the identity are constant. Also, the smallest simple congruence \(\theta\) on such semigroups is described. Necessary and sufficient conditions are found in order that the lattice of congruences C(S) be semi-modular, M- symmetric, modular or distributive: C(S) has any of these properties iff C(Y) and \(C(S_{\alpha})\) for all \(\alpha\in Y\) have the same property and all the linking homomorphisms are constant. Finally, complementation in C(S) is investigated, together with the Boolean case (see also K. Auinger, loc. cit.): C(S) is complemented iff S is a subdirect product of S/J and S/\(\theta\) and both C(S/J) and C(S/\(\theta)\) are complemented.