Tolerance trivial algebras and varieties. (English) Zbl 0534.08001

Chajda, Ivan

Acta Sci. Math. 46, 35-40 (1983).

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

MSC:

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

Keywords:

congruence permutable; principal tolerance triviality of varieties