Semigroup Forum

, Volume 85, Issue 3, pp 477-512

First online:

Open Access This content is freely available online to anyone, anywhere at any time.

On definability in some lattices of semigroup varieties

  • Mariusz GrechAffiliated withInstitute of Mathematics, University of Wrocław Email author 
  • , Olga SapirAffiliated withDepartment of Mathematics, Vanderbilt University


An identity of the form x 1x n x 1π x 2π x where π is a non-trivial permutation on the set {1,…,n} is called a permutation identity. If uv is a permutation identity, then (uv) [respectively r(uv)] 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 [respectively r] such that \(\mathcal{V}\) satisfies a permutation identity τ with (τ)= [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 ,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 r or \(\mathcal{V}\) is definable up to duality in \(L(\operatorname{var}{\mathfrak{B}}_{\ell,r,n})\) if =r.


Permutation identity Permutative variety 0-Reduced identity Nil-variety Lattice of subvarieties First order definability