Topological dualities and completions for (distributive) partially ordered sets

Resumen

This PhD thesis is the result of our research on duality theory and completions for partially ordered sets. A first main aim of this dissertation is to propose different kind of topological dualities for some classes of partially ordered sets and a second aim is to try to use these dualities to obtain completions with nice properties. To this end, we intend to follow the line of the classical dualities for bounded distributive lattices due to Stone and Priestley. Thus, we will need to consider a notion of distributivity on partially ordered sets. Also we propose a topological duality for the class of all partially ordered sets and we use this duality to study some properties of partially ordered sets like its canonical extension, order-preserving maps and the extensions of n-ary maps that are order-preserving in each coordinate. Moreover, to attain these aims we will study the partially ordered sets from an algebraic point of view.