Finite basis problem for Lee monoids with involution

Abstract

For each k >= 2, let L-k(1) denote the monoid obtained by adjoining an identity element to the semigroup generated by two idempotents E and F subjected to the relation EFEF...=0, where the left side is a product of length k. Under a certain unary operation *, the unary monoid (L-k(1), *) is an involution monoid. By recent results, the monoid L-k(1) is non-finitely based if and only if k >= 3. As for the involution monoid (L-k(1), *), it is only known to be non-finitely based for all even k >= 2. The objective of this article is to show that (L-k(1), *) is non-finitely based for all odd k >= 3. Consequently, (L-k(1), *) is non-finitely based for all k >= 2.