Relational representations for bounded lattices with operators

Goosen, Gerrit (2010-03)

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010.


ENGLISH ABSTRACT: Within lattice theory, an interesting question asked is whether a given abstract lattice may be represented concretely as subsets of a closure system on a topological space. This is true for boolean algebras, bounded distributive lattices and arbitrary bounded lattices. In particular, there are a multitude of ways to represent bounded lattices. We present some of these ideas, as well as an analysis of the differences between them. We further investigate the attempts that were made to extend the above representations to lattices endowed with operators, in particular the work done on bounded distributive lattices with operators. We then make a new contribution by extending this work to arbitrary bounded lattices with operators. We also show that the so-called sufficiency operator has a relational representation in the bounded lattice case.

AFRIKAANSE OPSOMMING: Binne die raamwerk van tralie teorie word die vraag soms gevra of ’n gegewe tralie konkreet veteenwoordig kan word as subversamelings van ’n afsluitingssisteem op ’n topologiese ruimte. Die voorgenoemde is waar vir, onder andere, boolse algebras, begrensde distributiewe tralies en algemene begrensde tralies. Daar is veral vir begrensde tralies menigte maniere om hul te verteenwoordig. Ons bied sommige van hierdie idees voor, asook ’n analiese van die verskille daarin teenwoordig. Verder ondersoek ons ook sommige van die maniere waarop tralies tesame met operatore verteenwoordig kan word. Ons sal spesiale aandag gee aan distributiewe tralies met operatore, soos gedoen in, met die idee om die voorgenoemde uit te brei na algemene begrensde tralies met operatore. Ons toon dan verder aan dat die sogenoemde voldoende operator ook ’n relasionele verteenwoordiging het in die begrensde tralie geval.

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