Algebras with principal tolerances. (English) Zbl 0617.08012

A tolerance on an algebra A is a reflexive and symmetric binary relation on A having the substitution property with respect to the operations of A. An algebra is called tolerance principal if each of its finitely generated tolerances can be generated by one pair of elements, and a variety of algebras is termed tolerance principal if each of its members has this property. The author proves a polynomial characterization of tolerance principal varieties; as an example, he notes a peculiar variety of groupoids. A properly modified version of tolerance principalness for algebras with constants admits varieties of lattices with 0 or 1 as examples.
Reviewer: M.Armbrust


08B10 Congruence modularity, congruence distributivity
08A30 Subalgebras, congruence relations
08B15 Lattices of varieties
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[1] CHAJDA I.: Tolerance trivial algebras and varieties. Acta Sci. Math. (Szeged), 46, 1983, 35-40. · Zbl 0534.08001
[2] CHAJDA I.: Distributivity and modulartiy of lattices of tolerance relations. Algebra Universalis 12, 1981, 247-255. · Zbl 0469.08003 · doi:10.1007/BF02483883
[3] CHAJDA I.: A Maľcev condition for congruence principal permutable varieties. Algebra Univ. 19, 1984, 337-340. · Zbl 0552.08006 · doi:10.1007/BF01201102
[4] CHAJDA I., ZELINKA B.: Minimal compatible tolerances on lattices. Czech, Math. J., 27, 1977, 452-459. · Zbl 0379.06002
[5] CHAJDA I., ZELINKA B.: Lattices of tolerances. Časop. pěst. matem. 102, 1977, 10-24. · Zbl 0354.08011
[6] NIEDERLE J.: Conditions for trivial principal tolerances. Archiv. Math. (Brno), 19, 1983, 145-152. · Zbl 0538.08002
[7] QUACKENBUSH R. W.: Varieties with n-principal compact congruences. Algebra Univ. 14, 1982, 292-296. · Zbl 0493.08006 · doi:10.1007/BF02483933
[8] ZLATOŠ P.: A Maľcev condition for compact congruences to be principal. Acta Sci. Math. (Szeged), 43, 1981, 383-387.
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