On definability in some lattices of semigroup varieties
 Mariusz Grech,
 Olga Sapir
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Abstract
An identity of the form x _{1}⋯x _{ n }≈x _{1π } x _{2π }⋯x _{ nπ } where π is a nontrivial 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 nilsemigroups is called a nilvariety. 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 nilvariety \(\mathcal{V}\) 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 firstorder 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.
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 Title
 On definability in some lattices of semigroup varieties
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Semigroup Forum
Volume 85, Issue 3 , pp 477512
 Cover Date
 20121201
 DOI
 10.1007/s0023301294298
 Print ISSN
 00371912
 Online ISSN
 14322137
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Permutation identity
 Permutative variety
 0Reduced identity
 Nilvariety
 Lattice of subvarieties
 First order definability
 Authors

 Mariusz Grech ^{(1)}
 Olga Sapir ^{(2)}
 Author Affiliations

 1. Institute of Mathematics, University of Wrocław, pl. Grunwaldzki 2, 50384, Wrocław, Poland
 2. Department of Mathematics, Vanderbilt University, Nashville, TN, 37240, USA