On conjugacy in regular epigroups

Let $S$ be a semigroup. The elements $a,b\in S$ are called primarily conjugate if $a=xy$ and $b=yx$ for certain $x,y\in S$. The relation of conjugacy is defined as the transitive closure of the relation of primary conjugacy. In the case when $S$ is a monoid, denote by $G$ the group of units of $S$. Then the relation of $G$-conjugacy is defined by $a\sim_G b \iff a=g^{-1}bg$ for certain $g\in G$. We establish the structure of conjugacy classes for regular epigroups (i.e. semigroups such that some power of each element lies in a subgroup). As a corollary we obtain a criterion of conjugacy in terms of $G$-conjugacy for factorizable inverse epigroups. We show that our general conjugacy criteria easily lead to known and new conjugacy criteria for some specific semigroups, among which are the full transformation semigroup and the full inverse symmetric semigroup over a finite set, the linear analogues of these semigroups and the semigroup of finitary partial automatic transformations over a finite alphabet.