## Modularity and distributivity of tolerance lattices of commutative inverse semigroups.(English)Zbl 0581.20058

Since the problem of characterizing all semigroups in certain classes with a given type of congruence lattice has attracted wide attention providing numerous satisfying results, the same problem is now investigated for tolerance lattices. A tolerance on a semigroup S is a reflexive and symmetric subalgebra of $$S\times S$$, i.e. a congruence relation on S which is not transitive. This paper studies necessary and sufficient conditions for a commutative inverse semigroup to have a modular or distributive lattice of tolerances. In a series of 16 Lemmas a characterization expressed mainly by order theoretical properties of idempotents is given in both cases. The corresponding results for congruence lattices of (arbitrary) Clifford semigroups were found by J. B. Fountain and P. Lockley [Semigroup Forum 14, 81-91 (1977; Zbl 0392.20041)]. As a consequence it is shown that for a semilattice S modularity and distributivity of the tolerance lattice of S are equivalent (as it is in the case of congruence lattices).
Reviewer: H.Mitsch

### MSC:

 20M10 General structure theory for semigroups 20M14 Commutative semigroups 06B15 Representation theory of lattices 20M15 Mappings of semigroups 08A30 Subalgebras, congruence relations

Zbl 0392.20041

### References:

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