Independent joins of tolerance factorable varieties

TL;DR

This paper demonstrates that the property of strong tolerance factorability in algebraic varieties is preserved under independent joins, allowing the construction of infinitely many such varieties with proper tolerances.

Contribution

It proves that strong tolerance factorability is maintained when forming independent joins of varieties, expanding the class of known strongly tolerance factorable varieties.

Findings

Strong tolerance factorability is preserved under independent joins.

Infinitely many strongly tolerance factorable varieties with proper tolerances are constructed.

Tolerances in such varieties are exactly the homomorphic images of congruences.

Abstract

Let L denote the variety of lattices. In 1982, the second author proved that L is strongly tolerance factorable, that is, the members of L have quotients in L modulo tolerances, although L has proper tolerances. We did not know any other nontrivial example of a strongly tolerance factorable variety. Now we prove that this property is preserved by forming independent joins (also called products) of varieties. This enables us to present infinitely many {strongly} tolerance factorable varieties with proper tolerances. Extending a recent result of G.\ Cz\'edli and G.\ Gr\"atzer, we show that if V is a strongly tolerance factorable variety, then the tolerances of V are exactly the homomorphic images of congruences of algebras in V. Our observation that (strong) tolerance factorability is not necessarily preserved when passing from a variety to an equivalent one leads to an open problem.