The structure and definability in the lattice of equational theories of strongly permutative semigroups
Mariusz Grech · Transactions of the American Mathematical Society · 2012
In this paper, we study the structure and the first-order definability in the lattice L ( S P ) \mathcal L(SP) of equational theories of strongly permutative semigroups, that is, semigroups satisfying a permutation identity \[ x 1 ⋯ x n = x σ ( 1 ) ⋯ x σ ( n ) x_1 \cdots x_n = x_{\sigma (1)} \cdots x_{\sigma (n)} \] with σ ( 1 ) > 1 \sigma (1) > 1 and σ ( n ) > n \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 L ( S P ) \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.