Direct finiteness of idempotents, primitivity and stability in semigroups

An idempotent [Formula: see text] in a semigroup [Formula: see text] is directly finite if, for any [Formula: see text], [Formula: see text] implies [Formula: see text]. In this paper, we systematically study directly finite idempotents in semigroups. First, we characterize them in terms of semigroup and [Formula: see text]-act properties. Then, we explore more precisely the relationship between directly finite idempotents and the stability of regular elements. We notably prove that a regular [Formula: see text]-class contains a directly finite idempotent if and only if its (regular) elements are stable. This result leads to the study of new Morita invariant properties for semigroups with local units: complete and pure direct finiteness, and complete and pure direct infiniteness. Additionally, we discuss applications to generalized inverses, intra-regular and [Formula: see text]-chained semigroups, and IC rings.