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.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:sun/oai:scholar.sun.ac.za:10019.1/4343 |
| Date | 03 1900 |
| Creators | Goosen, Gerrit |
| Contributors | Rewitzky, Ingrid, University of Stellenbosch. Faculty of Science. Dept. of Mathematical Sciences. |
| Publisher | Stellenbosch : University of Stellenbosch |
| Source Sets | South African National ETD Portal |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 70 p. |
| Rights | University of Stellenbosch |
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