Solid set

In mathematics, specifically in order theory and functional analysis, a subset ${\displaystyle S}$ of a vector lattice ${\displaystyle X}$ is said to be solid and is called an ideal if for all ${\displaystyle s\in S}$ and ${\displaystyle x\in X,}$ if ${\displaystyle |x|\leq |s|}$ then ${\displaystyle x\in S.}$ An ordered vector space whose order is Archimedean is said to be Archimedean ordered.¹ If ${\displaystyle S\subseteq X}$ then the ideal generated by ${\displaystyle S}$ is the smallest ideal in ${\displaystyle X}$ containing ${\displaystyle S.}$ An ideal generated by a singleton set is called a principal ideal in ${\displaystyle X.}$

The intersection of an arbitrary collection of ideals in ${\displaystyle X}$ is again an ideal and furthermore, ${\displaystyle X}$ is clearly an ideal of itself; thus every subset of ${\displaystyle X}$ is contained in a unique smallest ideal.

In a locally convex vector lattice ${\displaystyle X,}$ the polar of every solid neighborhood of the origin is a solid subset of the continuous dual space ${\displaystyle X^{\prime }}$; moreover, the family of all solid equicontinuous subsets of ${\displaystyle X^{\prime }}$ is a fundamental family of equicontinuous sets, the polars (in bidual ${\displaystyle X^{\prime \prime }}$) form a neighborhood base of the origin for the natural topology on ${\displaystyle X^{\prime \prime }}$ (that is, the topology of uniform convergence on equicontinuous subset of ${\displaystyle X^{\prime }}$).²

• A solid subspace of a vector lattice ${\displaystyle X}$ is necessarily a sublattice of ${\displaystyle X.}$¹
• If ${\displaystyle N}$ is a solid subspace of a vector lattice ${\displaystyle X}$ then the quotient ${\displaystyle X/N}$ is a vector lattice (under the canonical order).¹