A weakening residuation in adjoining closures
Isabella Mastroeni, Roberto Giacobazzi · 2015
In this paper we weaken the conditions for the existence of adjoint closure operators, 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 closure 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.