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 ${\displaystyle \left[{\mathcal {F))\right]_{C}:=\left\{[F]_{C}:F\in {\mathcal {F))\right\}.}$