Inverse semigroups with apartness

The notion of semigroups with apartness has been introduced recently as a constructive counterpart of classical semigroups. On such structures, a constructive analogue of the isomorphism theorem has been proved, and quasiorder relations and related substructures have been studied. In this paper, we extend this approach by introducing inverse semigroups with apartness, a useful tool to describe partial symmetries in sets with apartness. We prove a constructive analogue of the isomorphism theorem for inverse semigroups and provide a characterisation of cocongruences on inverse semigroups.