A weakening residuation in adjoining closures

In this paper we weaken the conditions for the existence of adjoint closure opera- tors, going beyond the standard requirement of additivity/co-additivity. We move from the notion of join-uniform (lower) closure operators, introduced in computer science in order to model perfect lossless compression in transformations acting on complete lattices. Starting from Janowitz’s characterisation of residuated clo- sure operators, we show that join-uniformity perfectly weakens additivity in the construction of residuated closures, and this is indeed the weakest property for this to hold. We conclude by characterising the set of all join-uniform lower closure operators as fix-points of a function defined on the set of all lower closures of a complete lattice.