Definability in the lattice of equational theories of commutative semigroups
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Abstract:
In this paper we study first-order definability in the lattice of equational theories of commutative semigroups. In a series of papers, J. Ježek, solving problems posed by A. Tarski and R. McKenzie, has proved, in particular, 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). He has proved also that the most important classes of theories of a given type are so definable. In a later paper, Ježek and McKenzie have “almost proved" the same facts for the lattice of equational theories of semigroups. There were good reasons to believe that the same can be proved for the lattice of equational theories of commutative semigroups. In this paper, however, we show that the case of commutative semigroups is different.References
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Additional Information
- Andrzej Kisielewicz
- Affiliation: Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
- Email: kisiel@math.uni.wroc.pl
- Received by editor(s): June 14, 2002
- Received by editor(s) in revised form: March 21, 2003
- Published electronically: October 28, 2003
- Additional Notes: This research was done while the author was a Fulbright Visiting Scholar at Vanderbilt University. Supported in part by Polish KBN grant P03A 00916.
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 356 (2004), 3483-3504
- MSC (2000): Primary 03C07; Secondary 03C05, 08B15, 20M07
- DOI: https://doi.org/10.1090/S0002-9947-03-03351-8
- MathSciNet review: 2055743
Dedicated: To Professor Ralph McKenzie