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.

A weakening residuation in adjoining closures

MASTROENI, Isabella;GIACOBAZZI, Roberto
2015-01-01

Abstract

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.
2015
Residuated closures, uniformity, adjoint functions
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11562/925745
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