Automorphisms of the lattice of equational theories of commutative semigroups. (English) Zbl 1187.08003
The authors study first-order definability in the lattice of equational theories of commutative semigroups. The author describes the group of automorphisms of this lattice and solves related problems posed by R. McKenzie and A. Kisielewicz.
Reviewer: Jiří Rosický (Brno)
MSC:
08B15 | Lattices of varieties |
03C05 | Equational classes, universal algebra in model theory |
03E40 | Other aspects of forcing and Boolean-valued models |
References:
[1] | Mariusz Grech, Irreducible varieties of commutative semigroups, J. Algebra 261 (2003), no. 1, 207 – 228. · Zbl 1026.20040 · doi:10.1016/S0021-8693(02)00674-9 |
[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 · doi:10.1006/jabr.2000.8383 |
[3] | 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. |
[4] | 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. |
[5] | 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. |
[6] | 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 |
[7] | 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 · doi:10.1007/BF02573566 |
[8] | Andrzej Kisielewicz, Varieties of commutative semigroups, Trans. Amer. Math. Soc. 342 (1994), no. 1, 275 – 306. · Zbl 0801.20042 |
[9] | Andrzej Kisielewicz, Definability in the lattice of equational theories of commutative semigroups, Trans. Amer. Math. Soc. 356 (2004), no. 9, 3483 – 3504. · Zbl 1050.08005 |
[10] | Andrzej Kisielewicz, Permutability class of a semigroup, J. Algebra 226 (2000), no. 1, 295 – 310. · Zbl 0954.20031 · doi:10.1006/jabr.1999.8174 |
[11] | Ralph McKenzie, Definability in lattices of equational theories, Ann. Math. Logic 3 (1971), no. 2, 197 – 237. · Zbl 0328.02038 · doi:10.1016/0003-4843(71)90007-6 |
[12] | Peter Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298 – 314. · Zbl 0186.03401 · doi:10.1016/0021-8693(69)90058-1 |
[13] | A. Tarski, Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 275 – 288. |
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