Tolerance distributive and tolerance modular varieties of commutative semigroups. (English) Zbl 0614.20043

A “tolerance” on a semigroup \(S\) is a reflexive and symmetric subsemigroup of \(S\times S\). The author proves the following theorem: a variety \(V\) of commutative semigroups is tolerance modular, that is, each member of \(V\) has modular lattice of tolerances, if and only if \(V\) satisfies an identity \(xy=xyz^ n\) for some positive integer n. Further, amongst such varieties, only the variety of “zero”, or “null”, semigroups and the trivial variety are tolerance distributive.
Reviewer: P.R.Jones


20M07 Varieties and pseudovarieties of semigroups
08A30 Subalgebras, congruence relations
08B10 Congruence modularity, congruence distributivity
20M14 Commutative semigroups


[1] Chajda I.: Lattices of compatible relations. Arch. Math. (Brno) 13 (1977), 89-96. · Zbl 0372.08002
[2] Chajda I., Zelinka B.: Lattices of tolerances. Čas. pěst. mat. 102 (1977), 10-24. · Zbl 0354.08011
[3] Chajda I.: Distributivity and modularity of lattices of tolerance relations. Algebra Universalis 12 (1981), 247-255. · Zbl 0469.08003
[4] Clifford A. H., Preston G. B.: The algebraic theory of semigroups. Vol. I. Am. Math. Soc., 1961. · Zbl 0111.03403
[5] Petrich M.: Introduction to Semigroups. Merill Publishing Company, 1973. · Zbl 0321.20037
[6] Pondělíček B.: Modularity and distributivity of tolerance lattices of commutative separative semigroups. Czech. Math. J. 35 (1985), 333 -337. · Zbl 0573.20062
[7] Zelinka B.: Tolerance in algebraic structures II. Czech. Math. J. 25 (1975), 175-178. · Zbl 0316.08001
[8] Ore O.: Structures and group theory II. Duke Math. J. 4 (1938), 247-269. · Zbl 0020.34801
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