In mathematics, specifically in order theory and functional analysis, a subset
of a vector lattice
is said to be solid and is called an ideal if for all
An ordered vector space whose order is Archimedean is said to be Archimedean ordered.
An ideal generated by a singleton set is called a principal ideal in
The intersection of an arbitrary collection of ideals in
is contained in a unique smallest ideal.
In a locally convex vector lattice
the polar of every solid neighborhood of the origin 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 (in bidual
) form a neighborhood base of the origin for the natural topology on
(that is, the topology of uniform convergence on equicontinuous subset of