Solid set


In mathematics, specifically in order theory and functional analysis, a subset S of a vector lattice is said to be solid and is called an ideal if for all s in S and x in X, if |x| ≤ |s| then x belongs to S.
An ordered vector space whose order is Archimedean is said to be Archimedean ordered.
If S is a subset of X then the ideal generated by S is the smallest ideal in X containing S.
An ideal generated by a singleton set is called a principal ideal in X.

Examples

The intersection of an arbitrary collection of ideals in X is again an ideal and furthermore, X is clearly an ideal of itself;
thus every subset of X is contained in a unique smallest ideal.
In a locally convex vector lattice X, the polar of every solid neighborhood of 0 is a solid subset of the continuous dual space ;
moreover, the family of all solid equicontinuous subsets of is a fundamental family of equicontinuous sets, the polars form a neighborhood base of the origin for the natural topology on .

Properties