Finitely based, finite sets of words (Q2709973)

For a finite set \(W\) of words, \(S(W)\) stands for the monoid defined on the set consisting of all subwords of words in \(W\) and of \(0\) by letting the product of two subwords \(u\) and \(v\) be \(uv\) if \(uv\) is again a subword of a word in \(W\) and be \(0\) otherwise. This construction was suggested by \textit{P.~Perkins} [J. Algebra 11, 298-314 (1969; Zbl 0186.03401)], who used it to produce one of the first examples of a non-finitely based finite semigroup. The authors systematically study the finite basis problem for monoids of the form \(S(W)\). They find many interesting results showing that, in a sense, this class of monoids behaves with respect to the finite basis property as irregularly as the class of all finite semigroups. For example, with every finite set \(W\), one can start an infinite increasing sequence \(W=W_1\subset W_2\subset\dots\) of finite sets of words such that in the sequence \(S(W_k)\), \(k=1,2,\dots\), finitely and non-finitely based monoids alternate. One more example: there are finite sets \(V_1,V_2\) such that the monoids \(S(V_1)\) and \(S(V_2)\) are non-finitely based [finitely based], while their direct product \(S(V_1)\times S(V_2)\) is finitely based [respectively, non-finitely based].