A class of covers of the free monogenic inverse monoid is introduced in one‐to‐one correspondence with the irrational numbers greater than one. Each member of the class is an eventually regular semigroup with a generatora, the regular powers of which form a pseudo‐random set. This then affords an example of a direct product of two eventually regular semigroups that is not eventually regular, thus answering a question of P. M. Edwards.