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arXiv · 1207.1732

Independent joins of tolerance factorable varieties

Ivan Chajda·Gábor Czédli·Radomir Halas
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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.

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Ivan Chajda, Gábor Czédli, Radomir Halas. 2012-07-06. Independent joins of tolerance factorable varieties. https://doi.org/10.1007/s00012-012-0213-0

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