Cone-saturated

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

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

[math]\displaystyle{ C }[/math]-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.

Properties

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

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

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

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice
• Locally convex vector lattice

Bibliography

• Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. 
• Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.