Lattice Universal Semigroup Varieties

A semigroup variety $\mathcal V$ is called \emph{lattice universal} if its subvariety lattice $L(\mathcal V)$ contains an interval dual to the partition lattice on a countably infinite set. (In this case the subvariety lattice of every variety of algebras of at most countable similarity type embeds into $L(\mathcal V)$, and this justifies the name.) The first example of a lattice universal semigroup variety was given by Burris and Nelson in 1971: this is the variety defined by the identity $x^2=x^3$. In 1976 Je\v{z}ek showed that the variety of semigroups with 0 in which $x^2=0$ also is lattice universal. We have classified lattice universal varieties in a large class of varieties. Recall the definition of the so-called Zimin words $Z_n$: one sets $Z_1=x_1$ and $Z_{n+1}=Z_nx_{n+1}Z_n$ for $n=1,2,\dotsc$. \begin{theorem} Suppose that a semigroup variety $\mathcal V$ is defined by identities depending on at most $n$ variables and all periodic groups in $\mathcal V$ are locally finite. Then $\mathcal V$ is lattice universal if and only if it does not satisfy any non-trivial identity of the form $Z_{n+1}=w$. \end{theorem}