A categorical approach to lattice-like structures
Michael Hoefnagel · SUNScholar (Stellenbosch University) · 2018
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.