The Free Distributive Semilattice Extension of a Distributive Poset

摘要

The main aim of this paper is to obtain a free distributive semilattice extension for some ordered sets satisfying a distributivity condition. That is, for an ordered set P satisfying a distributivity condition, we prove the existence of a distributive semilattice M and a monomorphism e : P hooked right arrow M such that for every distributive semilattice L and every monomorphism f : P hooked right arrow L there exists a unique semilattice embedding degrees e. To attain this, we will need to consider and study some concepts on ordered sets like filters and a distributivity condition. We consider three notions of filters on posets known in the literature, and we show some new relationships between them. We also introduce and investigate three definitions of morphisms between posets.