Order complete

In mathematics, specifically in order theory and functional analysis, a subset $$A$$ of an ordered vector space is said to be order complete in $$X$$ if for every non-empty subset $$S$$ of $$X$$ that is order bounded in $$A$$ (meaning contained in an interval, which is a set of the form $$[a, b] := \{ x \in X : a \leq x \text{ and } x \leq b \},$$ for some $$a, b \in A$$), the supremum $$\sup S$$' and the infimum $$\inf S$$ both exist and are elements of $$A.$$ An ordered vector space is called order complete, Dedekind complete, a complete vector lattice, or a complete Riesz space, if it is order complete as a subset of itself, in which case it is necessarily a vector lattice. An ordered vector space is said to be countably order complete if each countable subset that is bounded above has a supremum.

Being an order complete vector space is an important property that is used frequently in the theory of topological vector lattices.

Examples
The order dual of a vector lattice is an order complete vector lattice under its canonical ordering.

If $$X$$ is a locally convex topological vector lattice then the strong dual $$X^{\prime}_b$$ is an order complete locally convex topological vector lattice under its canonical order.

Every reflexive locally convex topological vector lattice is order complete and a complete TVS.

Properties
If $$X$$ is an order complete vector lattice then for any subset $$S \subseteq X,$$ $$X$$ is the ordered direct sum of the band generated by $$A$$ and of the band $$A^{\perp}$$ of all elements that are disjoint from $$A.$$ For any subset $$A$$ of $$X,$$ the band generated by $$A$$ is $$A^{\perp \perp}.$$ If $$x$$ and $$y$$ are lattice disjoint then the band generated by $$\{x\},$$ contains $$y$$ and is lattice disjoint from the band generated by $$\{y\},$$ which contains $$x.$$