Abstract
Let Lat denote the variety of lattices. In 1982, the second author proved that Lat is strongly tolerance factorable, that is, the members of Lat have quotients in Lat modulo tolerances, although Lat 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 \({\mathcal{V}}\) is a strongly tolerance factorable variety, then the tolerances of \({\mathcal{V}}\) are exactly the homomorphic images of congruences of algebras in \({\mathcal{V}}\). 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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Presented by E. Kiss.
Dedicated to Béla Csákány on his eightieth birthday
This research was supported by the project Algebraic Methods in Quantum Logic, no. CZ.1.07/2.3.00/20.0051, under the NFSR of Hungary (OTKA), grant numbers K77432 and K83219, and by TÁMOP-4.2.1/B-09/1/KONV-2010-0005.
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Chajda, I., Czédli, G. & Halaš, R. Independent joins of tolerance factorable varieties. Algebra Univers. 69, 83–92 (2013). https://doi.org/10.1007/s00012-012-0213-0
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DOI: https://doi.org/10.1007/s00012-012-0213-0