## 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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### References:

 [1] M. Grech, Irreducible varieties of commutative semigroups, J. Algebra, to appear. · Zbl 1026.20040 [2] Mariusz Grech and Andrzej Kisielewicz, Covering relation for equational theories of commutative semigroups, J. Algebra 232 (2000), no. 2, 493 – 506. · Zbl 0970.20033 [3] Pierre Antoine Grillet, Fully invariant congruences on free commutative semigroups, Acta Sci. Math. (Szeged) 67 (2001), no. 3-4, 571 – 600. · Zbl 1004.20048 [4] A. A. Iskander, Coverings in the lattice of varieties, Contributions to universal algebra (Colloq., József Attila Univ., Szeged, 1975) North-Holland, Amsterdam, 1977, pp. 189 – 203. Colloq. Math. Soc. János Bolyai, Vol. 17. · Zbl 0375.08006 [5] Awad A. Iskander, Definability in the lattice of ring varieties, Pacific J. Math. 76 (1978), no. 1, 61 – 67. · Zbl 0403.08005 [6] Jaroslav Ježek, The lattice of equational theories. I. Modular elements, Czechoslovak Math. J. 31(106) (1981), no. 1, 127 – 152. With a loose Russian summary. Jaroslav Ježek, The lattice of equational theories. II. The lattice of full sets of terms, Czechoslovak Math. J. 31(106) (1981), no. 4, 573 – 603. Jaroslav Ježek, The lattice of equational theories. III. Definability and automorphisms, Czechoslovak Math. J. 32(107) (1982), no. 1, 129 – 164. [7] Jaroslav Ježek, The lattice of equational theories. I. Modular elements, Czechoslovak Math. J. 31(106) (1981), no. 1, 127 – 152. With a loose Russian summary. Jaroslav Ježek, The lattice of equational theories. II. The lattice of full sets of terms, Czechoslovak Math. J. 31(106) (1981), no. 4, 573 – 603. Jaroslav Ježek, The lattice of equational theories. III. Definability and automorphisms, Czechoslovak Math. J. 32(107) (1982), no. 1, 129 – 164. [8] Jaroslav Ježek, The lattice of equational theories. I. Modular elements, Czechoslovak Math. J. 31(106) (1981), no. 1, 127 – 152. With a loose Russian summary. Jaroslav Ježek, The lattice of equational theories. II. The lattice of full sets of terms, Czechoslovak Math. J. 31(106) (1981), no. 4, 573 – 603. Jaroslav Ježek, The lattice of equational theories. III. Definability and automorphisms, Czechoslovak Math. J. 32(107) (1982), no. 1, 129 – 164. [9] Jaroslav Ježek, The lattice of equational theories. IV. Equational theories of finite algebras, Czechoslovak Math. J. 36(111) (1986), no. 2, 331 – 341. · Zbl 0605.08005 [10] Jaroslav Ježek and Ralph McKenzie, Definability in the lattice of equational theories of semigroups, Semigroup Forum 46 (1993), no. 2, 199 – 245. · Zbl 0782.20051 [11] Andrzej Kisielewicz, Varieties of commutative semigroups, Trans. Amer. Math. Soc. 342 (1994), no. 1, 275 – 306. · Zbl 0801.20042 [12] Andrzej Kisielewicz, All pseudovarieties of commutative semigroups, Semigroups with applications (Oberwolfach, 1991) World Sci. Publ., River Edge, NJ, 1992, pp. 78 – 89. [13] Andrzej Kisielewicz, Unification in commutative semigroups, J. Algebra 200 (1998), no. 1, 246 – 257. · Zbl 0898.20037 [14] Ralph McKenzie, Definability in lattices of equational theories, Ann. Math. Logic 3 (1971), no. 2, 197 – 237. · Zbl 0328.02038 [15] Evelyn Nelson, The lattice of equational classes of commutative semigroups, Canad. J. Math. 23 (1971), 875 – 895. · Zbl 0226.08002 [16] Peter Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298 – 314. · Zbl 0186.03401 [17] A. Tarski, Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 275 – 288. [18] B. M. Vernikov, Definable varieties of associative rings, Mat. Issled. 90, Algebry i Kol$$^{\prime}$$tsa (1986), 41 – 47, 148 (Russian). · Zbl 0614.16012
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