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

Abstract

In this paper, we study the structure and the first-order definability in the lattice 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) with σ(1) > 1 and σ(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(SP); inclusion, operations ∨ and ∧, and covering relation. Using this description, we prove, in particular, that each individual theory of strongly permutative semigroups is definable, up to duality.

On definability in some lattices of semigroup varieties

“…a linear identity that is non-trivial modulo x(yz) ≈ (xy)z. In [3], the first-named author took the next step up from the case of COM to a more difficult case when the set B contains the associativity identity and some permutation identities.…”

“…We say that a lattice L of semigroup varieties is self-dual if L δ = L. If L is a self-dual sublattice of SEM then we say that a variety V ∈ L is semi-definable in L if the set {V, V δ } is definable in L. In [3], the first-named author proved that each 0-permutative variety is semidefinable in P(0) and consequently, the lattice P(0) has no nontrivial automorphisms except δ.…”

“…In particular, each 0-permutative variety is contained in L(var B 0,0,n ) for some n > 1. The most difficult part of proving [3,Theorem 10.9] that each 0-permutative variety is semi-definable in P(0) was proving that each 0-permutative variety is semi-definable in L(var B 0,0,n ) for some n big enough.…”

“…The same arguments can be used to show that the class of all nilvarieties in I is definable in I (see Lemma 2.4). By using an idea from [3], a result from [23] and some functions on varieties, we prove (Proposition 4.1) that a nilvariety V is b0-reduced if and only if for each prime d > 1, the variety V ∨ var C 0,d is a cover of the variety V. This implies (Theorem 4.1) that the set of all b0-reduced varieties in I is definable in I. This statement plays a similar role in our article as Theorem 1.11 in [11].…”

“…Based on the proof of [3,Theorem 9.3], we show (Corollaries 6.1 and 6.2) that for each , r ≥ 0 and n > 1 each word pattern is semi-definable in F ∞ /⇔ B ,r,n if = r and is definable in F ∞ /⇔ B ,r,n if = r. Corollaries 6.1, 6.2 and Theorem 8.1 imply (Theorem 9.1) that for each , r ≥ 0 and n > 1, each variety of the form var{u ≈ 0,…”