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 $$n\leqslant 2$$ . [ABSTRACT FROM AUTHOR]