Band (order theory)
In mathematics, specifically in order theory and functional analysis, a band in a vector lattice X is a subspace M of X that is solid and such that for all S ⊆ M such that x = sup S exists in X, we have x ∈ M.
The smallest band containing a subset S of X is called the band generated by S in X.
A band generated by a singleton set is called a principal band.Examples
For any subset S of a vector lattice X, the set of all elements of X disjoint from S is a band in X.
If is the usual space of real valued functions used to define Lps, then is countably order complete but in general is not order complete.
If N is the vector subspace of all -null functions then N is a solid subset of that is not a band.Properties
The intersection of an arbitrary family of bands in a vector lattice X is a band in X.
