On the finite basis problem for the monoids of partial extensive injective transformations

Let \(PEI_n (POEI_n)\) be the monoid of all partial (order-preserving) extensive and injective transformations over a chain of order n. We give a sufficient condition under which a semigroup is nonfinitely based and apply this condition to show that the monoid \(PEI_3 (POEI_3)\) is nonfinitely based. This together with the results of Edmunds and Goldberg gives a complete answer to the finite basis problem for the monoid \(PEI_n (POEI_n)\): the monoid \(PEI_n (POEI_n)\) is nonfinitely based if and only if \(n\geqslant 3\). Furthermore, it is shown that the monoid \(PEI_n (POEI_n)\) is hereditarily finitely based if and only if \(n\leqslant 2\).