Weak differences, weak BCK-algebras and applications to some partial orders on rings

Abstract This article introduces a weak difference on a poset as a generalization of a difference on a poset and deals with connections between weak differences and weak BCK-algebras. In particular, conditions under which a poset with a certain weak difference is a meet semilattice are provided. These conditions are then applied to some partial orders on rings, namely the weak right star order and the strong right star order on a right-strong Rickart ring, as well as the sharp order on a certain subset of an arbitrary unitary ring.