Since the 1980s, it has been known that the smallest non-finitely based semigroups are of order six.Surprisingly, for involution semigroups, a nonfinitely based example of order five was recently discovered.In this article, it is confirmed that every involution semigroup of order four is finitely based.Since every involution semigroup of order three or less is already known to be finitely based, it follows that the smallest non-finitely based involution semigroups are of order five.
Gao et al. (Mon,) studied this question.