Solid set

In mathematics, specifically in order theory and functional analysis, a subset ${\displaystyle S}$ of a vector lattice 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. 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 ${\displaystyle X.}$