Finitely Based, Finite Sets of Words

For W a finite set of words, we consider the Rees quotient of a free monoid with respect to the ideal consisting of all words that are not subwords of W . This monoid is denoted by S(W ). It is shown that for every finite set of words W , there are sets of words U ⊃ W and V ⊃ W such that the identities satisfied by S(V ) are finitely based and those of S(U) are not finitely based (regardless of the situation for S(W )). The first examples of finitely based (not finitely based) aperiodic finite semigroups whose direct product is not finitely based (finitely based) are presented and it is shown that every monoid of the form S(W ) with fewer than 9 elements is finitely based and that there is precisely one not finitely based 9 element example.