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 $$|x| \leq |s|$$ then $$x \in S.$$ An ordered vector space whose order is Archimedean is said to be Archimedean ordered. If $$S\subseteq X$$ 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 the origin is a solid subset of the continuous dual space $$X^{\prime}$$; moreover, the family of all solid equicontinuous subsets of $$X^{\prime}$$ is a fundamental family of equicontinuous sets, the polars (in bidual $$X^{\prime\prime}$$) form a neighborhood base of the origin for the natural topology on $$X^{\prime\prime}$$ (that is, the topology of uniform convergence on equicontinuous subset of $$X^{\prime}$$).

Properties

 * A solid subspace of a vector lattice $$X$$ is necessarily a sublattice of $$X.$$
 * If $$N$$ is a solid subspace of a vector lattice $$X$$ then the quotient $$X/N$$ is a vector lattice (under the canonical order).