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