A categorical approach to lattice-like structures

Hoefnagel, Michael Anton (2018-12)

Thesis (PhD)--Stellenbosch University, 2018.

Thesis

ENGLISH ABSTRACT : This thesis is a first step in a categorical approach to lattice-like structures. Its central notion, that of a majority category, relates to the category of lattices, in a similar way as Mal’tsev categories relate to the category of groups. This notion provides a context in which to establish categorical counterparts of various lattice-theoretic results. Surprisingly, many categories of a geometric nature naturally possess the dual property; namely, they are comajority categories. We show that several characterizations of varieties admitting a majority term, extend to characterizations of regular majority categories. These characterizations then show how majority categories relate to other well known notions in the literature, such as arithmetical and protoarithmetical categories. The most interesting results, from the point of view of the author, are those that concern decomposition and factorization. For example, every subobject of a finite product of objects in a regular majority category is uniquely determined by its two-fold projections – which can be seen as a certain subobject decomposition property. One of the main points of the thesis proves that in a regular majority category, every product of directly-irreducible objects is unique.

AFRIKAANSE OPSOMMING : Hierdie proefskrif is ’n eerste stap na ’n kategoriese benadering tot roostersoos strukture. Die sentrale begrip daarvan, dié van ’n meerderheidskategorie, het betrekking op die kategorie van roosters, op soortgelyke wyse soos Mal’tsev-kategorieë betrekking het op die kategorie van groepe. Hierdie idee bied ’n konteks waarin kategoriese eweknieë van verskillende roosterteoretiese resultate gevestig kan word. Baie kategorieë van ’n meetkundige aard het die dubbele eienskap; naamlik, hulle is (co)meerderheids kategorieë. Ons wys dat verskeie karakters van variëteite wat ’n meerderheidstermyn toelaat, uitbrei na karakterisering van gereelde meerderheidskategorieë. Hierdie karakterisering toon dan aan hoe meerderheidskategorieë verband hou met ander bekende begrippe in die literatuur, soos Arithmetical en protoarithmetical kategorieë. Die mees interessante resultate, uit die oogpunt van die skrywer, is dié wat ontbinding en faktorisering betref. Ons wys dat direkte produkte erken ’n sekere unieke faktorisering stelling soortgelyk aan die universele algebraïese teendeel.

Please refer to this item in SUNScholar by using the following persistent URL: http://hdl.handle.net/10019.1/104833

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