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

Grech, Mariusz

doi:10.1090/S0002-9947-2012-05386-4

The structure and definability in the lattice of equational theories of strongly permutative semigroups
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Trans. Amer. Math. Soc. 364 (2012), 2959-2985 Request permission

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 \[ x_1 \cdots x_n = x_{\sigma (1)} \cdots x_{\sigma (n)}\]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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