On definability in some lattices of semigroup varieties

Abstract

An identity of the form x 1⋯x n ≈x 1π x 2π ⋯x nπ where π is a non-trivial permutation on the set {1,…,n} is called a permutation identity. If u≈v is a permutation identity, then ℓ(u≈v) [respectively r(u≈v)] is the maximal length of the common prefix [suffix] of the words u and v. A variety that satisfies a permutation identity is called permutative. If \(\mathcal{V}\) is a permutative variety, then \(\ell=\ell(\mathcal{V})\) [respectively \(r=r(\mathcal{V})\)] is the least ℓ [respectively r] such that \(\mathcal{V}\) satisfies a permutation identity τ with ℓ(τ)=ℓ [respectively r(τ)=r]. A variety that consists of nil-semigroups is called a nil-variety. If Σ is a set of identities, then \(\operatorname {var}\varSigma\) denotes the variety of semigroups defined by Σ. If \(\mathcal{V}\) is a variety, then \(L (\mathcal{V})\) denotes the lattice of all subvarieties of \(\mathcal{V}\).For ℓ,r≥0 and n>1 let \(\mathfrak{B}_{\ell,r,n}\) denote the set that consists of n! identities of the form $$t_1\cdots t_\ell x_1x_2 \cdots x_n z_{1}\cdots z_{r}\approx t_1\cdots t_\ell x_{1\pi}x_{2\pi} \cdots x_{n\pi}z_{1}\cdots z_{r}, $$ where π is a permutation on the set {1,…,n}. We prove that for each permutative nil-variety \(\mathcal{V}\) and each \(\ell\ge\ell(\mathcal{V})\) and \(r\ge r(\mathcal{V})\) there exists n>1 such that \(\mathcal{V}\) is definable by a first-order formula in \(L(\operatorname{var}{\mathfrak{B}}_{l,r,n})\) if ℓ≠r or \(\mathcal{V}\) is definable up to duality in \(L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})\) if ℓ=r.