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Modularity and distributivity of tolerance lattices of commutative separative semigroups. (English) Zbl 0573.20062

By a tolerance lattice on an algebra A is meant the lattice of all tolerances on A ordered by set inclusion. The paper contains a characterization of commutative regular semigroups whose tolerance lattice is modular or distributive. The results are applied in semigroup theoretical investigations: Theorem 3. Let S be a commutative semigroup with modular tolerance lattice. S is regular iff S is separative. Corollary 3. Let S be a commutative semigroup. S is cancellative with modular (or distributive) tolerance lattice iff S is a group (or locally cyclic group, respectively).
Reviewer: I.Chajda

MSC:

20M14 Commutative semigroups
20M15 Mappings of semigroups
08A30 Subalgebras, congruence relations
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References:

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