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.
Reviewer: B.Zelinka


08A30 Subalgebras, congruence relations
08B05 Equational logic, Mal’tsev conditions