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.