Cone-saturated

In mathematics, specifically in order theory and functional analysis, if $$C$$ is a cone at 0 in a vector space $$X$$ such that $$0 \in C,$$ then a subset $$S \subseteq X$$ is said to be $$C$$-saturated if $$S = [S]_C,$$ where $$[S]_C := (S + C) \cap (S - C).$$ Given a subset $$S \subseteq X,$$ the $$C$$-saturated hull of $$S$$ is the smallest $$C$$-saturated subset of $$X$$ that contains $$S.$$ If $$\mathcal{F}$$ is a collection of subsets of $$X$$ then $$\left[ \mathcal{F} \right]_C := \left\{ [F]_C : F \in \mathcal{F} \right\}.$$

If $$\mathcal{T}$$ is a collection of subsets of $$X$$ and if $$\mathcal{F}$$ is a subset of $$\mathcal{T}$$ then $$\mathcal{F}$$ is a fundamental subfamily of $$\mathcal{T}$$ if every $$T \in \mathcal{T}$$ is contained as a subset of some element of $$\mathcal{F}.$$ If $$\mathcal{G}$$ is a family of subsets of a TVS $$X$$ then a cone $$C$$ in $$X$$ is called a $$\mathcal{G}$$-cone if $$\left\{ \overline{[G]_C} : G \in \mathcal{G} \right\}$$ is a fundamental subfamily of $$\mathcal{G}$$ and $$C$$ is a strict $$\mathcal{G}$$-cone if $$\left\{ [B]_C : B \in \mathcal{B} \right\}$$ is a fundamental subfamily of $$\mathcal{B}.$$

$$C$$-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Properties
If $$X$$ is an ordered vector space with positive cone $$C$$ then $$[S]_C = \bigcup \left\{ [x, y] : x, y \in S \right\}.$$

The map $$S \mapsto [S]_C$$ is increasing; that is, if $$R \subseteq S$$ then $$[R]_C \subseteq [S]_C.$$ If $$S$$ is convex then so is $$[S]_C.$$ When $$X$$ is considered as a vector field over $$\R,$$ then if $$S$$ is balanced then so is $$[S]_C.$$

If $$\mathcal{F}$$ is a filter base (resp. a filter) in $$X$$ then the same is true of $$\left[ \mathcal{F} \right]_C := \left\{ [ F ]_C : F \in \mathcal{F} \right\}.$$