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 ${ℱ}$ is a collection of subsets of $X$ then ${\left[{ℱ}\right]}_C:=\{[F{]}_C:F\in {ℱ}\}.$

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

$C$-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 $[S{]}_C=\bigcup \{[x,y]:x,y\in S\}.$

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 $\mathbb{ℝ},$ then if $S$ is balanced then so is $[S{]}_C.$

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

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.