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The structure and definability in the lattice of equational theories of strongly permutative semigroups


Author: Mariusz Grech
Journal: Trans. Amer. Math. Soc. 364 (2012), 2959-2985
MSC (2000): Primary 03C07; Secondary 03C05, 08B15
DOI: https://doi.org/10.1090/S0002-9947-2012-05386-4
Published electronically: January 26, 2012
MathSciNet review: 2888235
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Abstract: In this paper, we study the structure and the first-order definability in the lattice $ \mathcal L(SP)$ of equational theories of strongly permutative semigroups, that is, semigroups satisfying a permutation identity

$\displaystyle x_1 \cdots x_n = x_{\sigma (1)} \cdots x_{\sigma (n)}$

with $ \sigma (1) > 1$ and $ \sigma (n) < n$. We show that each equational theory of such semigroups is described by five objects: an order filter, an equivalence relation, and three integers. We fully describe the lattice $ \mathcal L(SP)$; inclusion, operations $ \vee $ and $ \wedge $, and covering relation. Using this description, we prove, in particular, that each individual theory of strongly permutative semigroups is definable, up to duality.

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Additional Information

Mariusz Grech
Affiliation: Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2, 50-384 Wrocław, Poland
Email: Mariusz.Grech@math.uni.wroc.pl

DOI: https://doi.org/10.1090/S0002-9947-2012-05386-4
Keywords: Strongly permutative semigroup, lattice, equational theories
Received by editor(s): May 26, 2010
Published electronically: January 26, 2012
Additional Notes: The author was suported in part by Polish KBN grant 4319/PB/JM/10.
Article copyright: © Copyright 2012 American Mathematical Society

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