## Tolerance distributive and tolerance modular varieties of commutative semigroups.(English)Zbl 0614.20043

A “tolerance” on a semigroup $$S$$ is a reflexive and symmetric subsemigroup of $$S\times S$$. The author proves the following theorem: a variety $$V$$ of commutative semigroups is tolerance modular, that is, each member of $$V$$ has modular lattice of tolerances, if and only if $$V$$ satisfies an identity $$xy=xyz^ n$$ for some positive integer n. Further, amongst such varieties, only the variety of “zero”, or “null”, semigroups and the trivial variety are tolerance distributive.
Reviewer: P.R.Jones

### MSC:

 20M07 Varieties and pseudovarieties of semigroups 08A30 Subalgebras, congruence relations 08B10 Congruence modularity, congruence distributivity 20M14 Commutative semigroups

### References:

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