The finite basis problem for words with at most two non-linear variables

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. We call a set of words Wfinitely based if the monoid S(W) is finitely based. We find a simple algorithm that recognizes finitely based words among words with at most two non-linear variables. We also describe syntactically all hereditary finitely based monoids of the form S(W).