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The congruence lattice of a Tully semigroup. (English) Zbl 0774.20038

The concept of a “Tully” semigroup generalizes a construction used by E. J. Tully [J. Aust. Math. Soc. 9, 239-245 (1969; Zbl 0185.048)] in describing those semigroups all of whose ideals are retract ideals and a similar construction occurring in the author’s earlier papers on semigroups whose congruence lattices satisfy such conditions as complementedness. It is closely related to the construction by G. Lallement and M. Petrich of what are now called “strict regular” semigroups. Ultimately it can be traced back to A. H. Clifford’s description of semilattices of groups. The parameters for a Tully semigroup are a semilattice \(X\), a pairwise disjoint family of nontrivial 0-simple semigroups \(I_ \alpha\), \(\alpha\in X\), and for \(\alpha\geq\beta\) a partial homomorphism \(f_{\alpha,\beta}:I^*_ \alpha\to I^*_ \beta\). The compatibility conditions and multiplication (on the union of the \(I_ \alpha^*)\) are fairly simple generalizations of those in the “strong semilattice” construction. The corresponding semigroup is denoted \((X;I_ \alpha,f_{\alpha,\beta})\). The semigroups \(I_ \alpha\) turn out to be its principal factors.
The congruences on such a semigroup \(S\) are described in terms of the parameters of the construction. First, a congruence \(\rho\) on \(S\) induces a congruence \(\rho_{\mathcal J}\) on \(X\). With each element \(\overline\alpha\) of \(X/\rho_{\mathcal J}\) there is associated a 0-simple semigroup \(I_{\overline\alpha}\) on which \(\rho\) induces a congruence \(\rho_{\overline\alpha}\). The collection \((\rho_{\mathcal J};\rho_{\overline\alpha},\overline\alpha\in X/\rho_{\mathcal J})\) uniquely determines \(\rho\). Such “congruence aggregates” are abstractly characterized and joins and meets in the congruence lattice \({\mathcal C}(S)\) of \(S\) are derived in these terms. Finally, necessary and sufficient conditions are found in order that \({\mathcal C}(S)\), where \(S\) is Tully, be modular. These restrictions reduce, in certain special cases, to results of the reviewer and of M. Petrich on congruence lattices of strong semilattices of simple semigroups.

MSC:

20M10 General structure theory for semigroups
08A30 Subalgebras, congruence relations

Citations:

Zbl 0185.048
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References:

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