From congruence identities to tolerance identities
Paolo Lipparini · 2007
Abstract. We show, under a weak assumption on the term p, that a va-riety of general algebras satisfies the congruence identity p(α1,..., αn) ⊆ q(α1,..., αn) if and only if it satisfies the tolerance identity p(Θ1,...,Θn) ⊆ q(Θ1,...,Θn), provided we restrict ourselves to tolerances representable as R ◦ R−. Varieties in which every tolerance is representable include all congruence permutable varieties and all varieties of lattices. For arbitrary tolerances, the congruence identity p(α1,..., αn) ⊆ q(α1,..., αn) is equivalent to the identity p(Θ1 ◦ Θ1,...,Θn ◦ Θn) ⊆ q(Θ1 ◦Θ1,...,Θn ◦Θn). See Theorems 3.1, 3.2 and 3.3. Our arguments essentially deal with labeled graphs, rather than terms; we try to clarify the role of graphs in the study of Mal’cev conditions (see especially Proposition 7.6 and Theorem 7.7). 1.