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Definability in the lattice of equational theories of commutative semigroups. (English) Zbl 1050.08005

In a series of papers [Czech. Math. J. 31, 127–152 (1981; Zbl 0477.08006); ibid. 31, 573–603 (1981; Zbl 0486.08009); ibid. 32, 129–165 (1982; Zbl 0499.08005); ibid. 36, 331–341 (1986; Zbl 0605.08005)], J. Ježek solved some problems posed by A. Tarski and R. McKenzie about the problems of first-order definability in a lattice \(L\) of equational theories. In particular he proved that each equational theory is first-order definable in the lattice of equational theories of a given type, up to automorphism, and that such lattices have no automorphisms besides the obvious syntactically defined ones (with exceptions for special unary types). Also, he has proved that the most important classes of theories of a given type are so definable. In another paper, J. Ježek and R. McKenzie [Semigroup Forum 46, 199–245 (1993; Zbl 0782.20051)] have “almost proved” the same facts for the lattice of equational theories of semigroups. In this paper the author studies first-order definability in the lattice of equational theories of commutative semigroups. There were good reasons to believe that the same results as in the paper of J. Ježek and McKenzie can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, the author shows that the case of commutative semigroups is different.

MSC:

08B15 Lattices of varieties
03C05 Equational classes, universal algebra in model theory
20M07 Varieties and pseudovarieties of semigroups
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