Regularly ordered

In mathematics, specifically in order theory and functional analysis, an ordered vector space $$X$$ is said to be regularly ordered and its order is called regular if $$X$$ is Archimedean ordered and the order dual of $$X$$ distinguishes points in $$X$$. Being a regularly ordered vector space is an important property in the theory of topological vector lattices.

Examples
Every ordered locally convex space is regularly ordered. The canonical orderings of subspaces, products, and direct sums of regularly ordered vector spaces are again regularly ordered.

Properties
If $$X$$ is a regularly ordered vector lattice then the order topology on $$X$$ is the finest topology on $$X$$ making $$X$$ into a locally convex topological vector lattice.