Independent joins of tolerance factorable varieties

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.\ Czédli and G.\ Grätzer, 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.

Figures (1)

• Figure 1: $L$ and the blocks of $T$

Theorems & Definitions (8)

• Proposition 1: W. Taylor taylor, G. Grätzer, H. Lakser, and J. Pł onka gglakplonka
• Proposition 2
• Theorem 3
• Example 4
• Example 5
• Example 6
• Example 7
• Lemma 9