On the theory of epigroups. I

Epigroups are viewed as unary semigroups with a pseudoinverse operation. Criteria are found for the existence of a partition of an epigroup into unipotent subepigroups, the inheritability of this property by all homomorphic images of an epigroup, and the decomposability of an epigroup into a band (semilattice) of Archimedean epigroups. Here, and in the second part of this paper to follow, the focus is on characterizations in terms of 'prohibited' objects, mainly epifactors, i.e., homomorphic images of subepigroups; attention is also paid to characterizations of the epigroups under consideration in the language of identities, and possible prospects for future investigations on varieties of epigroups are outlined.