The notion of a pseudo-complement is extended to a broad class of semigroups, which we say are semilattice pseudo-complemented (SP). We examine the relationship between these and some other classes of unary semigroups, namely semigroups with closure and interior operations, and we also consider the concepts in a ring theoretic setting. Under some natural strengthenings of the defining axioms, some fundamental congruences are described, extending cases from the theory of inverse semigroups. The class of SP-semilattices coincides with a natural class of ordered structures related to topological spaces but differs slightly from the standard definition of a pseudo-complemented semilattice. These SP-semilattices arise naturally in the general theory and are given particular attention.