Non-finitely based finite involution semigroups with finitely based semigroup reducts

Recently, an infinite class of finitely based finite involution semigroups with non-finitely based semigroup reducts have been found. In contrast, only one example of the opposite type—non-finitely based finite involution semigroups with finitely based semigroup reducts—has so far been published. In the present article, a sufficient condition is established under which an involution semigroup is non-finitely based. This result is then applied to exhibit several examples of the desired opposite type.