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