The semigroup of weak endomorphisms of a partial equivalence relation

We study the weak endomorphism semigroup of a partial equivalence relation and describe necessary and sufficient conditions for the existence of these endomorphisms and, in the general case, endotopisms. We establish the conditions under which the set of all idempotents of the weak endomorphism semigroup of a strict partial equivalence is its subsemigroup, as well as the conditions of regularity and coregularity of the weak endomorphism semigroup. In terms of the wreath product of a symmetric transformation semigroup and a small category, we describe a faithful representation of the weak endomorphism semigroup of an arbitrary strict partial equivalence.