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In mathematics, specifically in order theory and functional analysis, if is a cone at 0 in a vector space such that then a subset is said to be **-saturated** if where
Given a subset the **-saturated hull** of is the smallest -saturated subset of that contains ^{[1]}
If is a collection of subsets of then

If is a collection of subsets of and if is a subset of then is a **fundamental subfamily** of if every is contained as a subset of some element of
If is a family of subsets of a TVS then a cone in is called a **-cone** if is a fundamental subfamily of and is a **strict -cone** if is a fundamental subfamily of ^{[1]}

-saturated sets play an important role in the theory of ordered topological vector spaces and topological vector lattices.

If is an ordered vector space with positive cone then ^{[1]}

The map is increasing; that is, if then
If is convex then so is When is considered as a vector field over then if is balanced then so is ^{[1]}

If is a filter base (resp. a filter) in then the same is true of