Solid set


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 and if then
An ordered [vector space] whose order is Archimedean is said to be Archimedean ordered.
If then the ideal generated by is the smallest ideal in containing
An ideal generated by a singleton set is called a principal ideal in

Examples

The intersection of an arbitrary collection of ideals in is again an ideal and furthermore, is clearly an ideal of itself;
thus every subset of 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 form a neighborhood base of the origin for the natural topology on .

Properties

  • A solid subspace of a vector lattice is necessarily a sublattice of
  • If is a solid subspace of a vector lattice then the quotient is a vector lattice.