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

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 {\em finitely based} if the monoid \(S(W)\) is finitely based. A word \(\bf u\) is called 2-limited if each variable occurs in \(\bf u\) at most twice. A {\em block} of a word \(\bf u\) is a maximal subword of \(\bf u\) that does not contain any linear variables. We say that a word \(\bf u\) is {\em block-2-simple} if each block of \(\bf u\) 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.