A categorical approach to lattice-like structures

[摘要] 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 ofa geometric nature naturally possess the dual property; namely, they arecomajority categories. We show that several characterizations of varietiesadmitting a majority term, extend to characterizations of regular majoritycategories. These characterizations then show how majority categories relate to other well known notions in the literature, such as arithmetical andprotoarithmetical categories. The most interesting results, from the point ofview 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 – whichcan be seen as a certain subobject decomposition property. One of the mainpoints of the thesis proves that in a regular majority category, every productof directly-irreducible objects is unique.

[发布机构] Stellenbosch University