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Transactions of the American Mathematical Society

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Definability in the lattice of equational theories of commutative semigroups


Author: Andrzej Kisielewicz
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
Published electronically: October 28, 2003
MathSciNet review: 2055743
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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. Jezek, 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, Jezek 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.


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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

DOI: https://doi.org/10.1090/S0002-9947-03-03351-8
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.
Dedicated: To Professor Ralph McKenzie
Article copyright: © Copyright 2003 American Mathematical Society