Finitely based sets of 2-limited block-2-simple words

Abstract

Let ${\mathfrak {A}}$ be an alphabet and W be a set of words in the free monoid ${{\mathfrak {A}}}^*$. Let S(W) denote the Rees quotient over the ideal of ${{\mathfrak {A}}}^*$ consisting of all words that are not subwords of words in W. A set of words W is called finitely based if the monoid S(W) is finitely based. A word $\mathbf{u}$ is called 2-limited if each variable occurs in $\mathbf{u}$ at most twice. A block of a word $\mathbf{u}$ is a maximal subword of $\mathbf{u}$ that does not contain any linear variables. A word $\mathbf{u}$ is block-2-simple if each block of $\mathbf{u}$ involves at most two distinct variables. We provide an algorithm that recognizes finitely based sets of words among sets of 2-limited block-2-simple words. We also present new sufficient conditions under which a set of words is non-finitely based.