Quasilinear varieties of semigroups. (Q2655452)

A semigroup variety \(\mathcal V\) is called `quasilinear' if for each word \(w\) there is a linear word \(w'\) such that \(\mathcal V\) satisfies \(w\approx w'\). If for each word \(w\) we have a unique \(w'\) then we obtain the notion of a `linear' variety of semigroups. In the paper there are characterized all quasilinear varieties of semigroups. There are precisely 9 quasilinear band varieties, namely the subvarieties of left regular bands \(\mathcal{LRB}\), right regular bands \(\mathcal{RRB}\) and rectangular bands \(\mathcal{RB}\). Let \(\mathcal V_n\) be the semigroup variety defined by the identities \(x^2\approx xyx\approx xy_1\cdots y_n\), while \(\mathcal V^d_n\) is the dual variety. A nonidempotent semigroup variety \(\mathcal V\) is quasilinear if and only if either \(\mathcal V\leq\mathcal V_n\) or \(\mathcal V\leq\mathcal V^d_n\) for some \(n\geq 1\). As a corollary we have the following assertion. There are precisely two linear semigroup varieties: \(\mathcal{LRB}\) and \(\mathcal{RRB}\).