## Tolerance trivial algebras and varieties.(English)Zbl 0534.08001

A tolerance on an algebra is defined similarly as a congruence, only the requirement of transitivity is omitted. A principal tolerance on an algebra $${\mathfrak A}$$ is the intersection of all tolerances on $${\mathfrak A}$$ which contain a given pair of distinct elements. An algebra is tolerance trivial (or principal tolerance trivial), if every tolerance (or every principal tolerance respectively) on $${\mathfrak A}$$ is a congruence. A variety $${\mathcal V}$$ of algebras is tolerance trivial or principal tolerance trivial, if every algebra of $${\mathcal V}$$ has the corresponding property. The first theorem states that a variety $${\mathcal V}$$ is tolerance trivial if and only if it is congruence permutable. Further theorems present various conditions for tolerance triviality and principal tolerance triviality of varieties.