Cone-saturated

In mathematics, specifically in order theory and functional analysis, if ${\displaystyle C}$ is a cone at 0 in a vector space ${\displaystyle X}$ such that ${\displaystyle 0\in C,}$ then a subset ${\displaystyle S\subseteq X}$ is said to be ${\displaystyle C}$-saturated if ${\displaystyle S=[S]_{C},}$ where ${\displaystyle [S]_{C}:=(S+C)\cap (S-C).}$ Given a subset ${\displaystyle S\subseteq X,}$ the ${\displaystyle C}$-saturated hull of ${\displaystyle S}$ is the smallest ${\displaystyle C}$-saturated subset of ${\displaystyle X}$ that contains ${\displaystyle S.}$^[1] If ${\displaystyle {\mathcal {F}}}$ is a collection of subsets of ${\displaystyle X}$ then ${\displaystyle \left[{\mathcal {F}}\right]_{C}:=\left\{[F]_{C}:F\in {\mathcal {F}}\right\}.}$

If ${\displaystyle {\mathcal {T}}}$ is a collection of subsets of ${\displaystyle X}$ and if ${\displaystyle {\mathcal {F}}}$ is a subset of ${\displaystyle {\mathcal {T}}}$ then ${\displaystyle {\mathcal {F}}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {T}}}$ if every ${\displaystyle T\in {\mathcal {T}}}$ is contained as a subset of some element of ${\displaystyle {\mathcal {F}}.}$ If ${\displaystyle {\mathcal {G}}}$ is a family of subsets of a TVS ${\displaystyle X}$ then a cone ${\displaystyle C}$ in ${\displaystyle X}$ is called a ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{{\overline {[G]_{C}}}:G\in {\mathcal {G}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {G}}}$ and ${\displaystyle C}$ is a strict ${\displaystyle {\mathcal {G}}}$-cone if ${\displaystyle \left\{[B]_{C}:B\in {\mathcal {B}}\right\}}$ is a fundamental subfamily of ${\displaystyle {\mathcal {B}}.}$^[1]

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

Properties

If ${\displaystyle X}$ is an ordered vector space with positive cone ${\displaystyle C}$ then ${\displaystyle [S]_{C}=\bigcup \left\{[x,y]:x,y\in S\right\}.}$^[1]

The map ${\displaystyle S\mapsto [S]_{C}}$ is increasing; that is, if ${\displaystyle R\subseteq S}$ then ${\displaystyle [R]_{C}\subseteq [S]_{C}.}$ If ${\displaystyle S}$ is convex then so is ${\displaystyle [S]_{C}.}$ When ${\displaystyle X}$ is considered as a vector field over ${\displaystyle \mathbb {R} ,}$ then if ${\displaystyle S}$ is balanced then so is ${\displaystyle [S]_{C}.}$^[1]

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

See also

• Banach lattice – Banach space with a compatible structure of a lattice
• Fréchet lattice – Topological vector lattice
• Locally convex vector lattice
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets

References

1. Schaefer & Wolff 1999, pp. 215–222.

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. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.