# Binary closure operators

Abdalla, Abdurahman Masoud (2016-03)

Thesis (PhD)--Stellenbosch University, 2016

Thesis

ENGLISH ABSTRACT : In this thesis we provide a new foundation to categorical closure operators, using more elementary binary closure operators on posets. The original goal of the thesis was to study a categorical closure operator in terms of the family of closure operators on the posets of subobjects. However, this does not allow to express hereditariness, which is an important property of a categorical closure operator. Representing instead a categorical closure operator in terms of the family of binary closure operators on the posets of subobjects, xes this problem. Moreover, the structure of a binary closure operator on a poset is self-dual, unlike that of a unary closure operator or that of a categorical closure operator, and this duality has a useful application in the study of properties of closure operators on categories, where it groups properties of categorical closure operators in dual pairs, and allows to unify results which relate these properties to each other.

AFRIKAANSE OPSOMMING : In hierdie tesis verskaf ons, deur gebruik te maak van meer elementêre binêre afsluitingsoperatore op parsiële geordende versamelings, 'n nuwe grondslag tot kategoriese afsluitingsoperatore. Die aanvanklike doel van die tesis was om 'n kategoriese afsluitingsoperator in terme van die familie van afsluitingsoperatore op parsiële die geordende versamelings van subobjekte te bestudeer. Dit laat egter nie toe om oorer ikheid, wat 'n belangrike eienskap van kategoriese operatore is, uit te druk nie. Hierdie probleem word opgelos deur 'n kategoriese operator in terme van die familie van binêre afsluitingsoperatore op parsiële die geordende versamelings van subobjekte te verteenwoordig. Bykomend is die struktuur van 'n binêre afsluitingsoperator op 'n parsiële geordende versameling self-duaal, in teenstelling met di e van 'n unêre of kategoriese afsluitingsoperator. Hierdie dualiteit het 'n nuttige toepassing in die studie van eienskappe van afsluitingsoperatore op kategorieë, waar dit eienskappe van kategoriese afsluitingsoperatore in duale pare groepeer en toelaat dat resultate, wat hierdie eienskappe in verband hou met mekaar, verenig word.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/98843

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