Band (order theory)
# 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 $S\subseteq M$ such that $x=\sup S$ exists in $X,$ we have $x\in M.$ 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 $S^{\perp }$ of all elements of $X$ disjoint from $S$ is a band in $X.$

If ${\mathcal {L}}^{p}(\mu )$ ($1\leq p\leq \infty$) is the usual space of real valued functions used to define Lp spaces $L^{p},$ then ${\mathcal {L}}^{p}(\mu )$ is countably order complete (that is, each subset that is bounded above has a supremum) but in general is not order complete. If $N$ is the vector subspace of all $\mu$-null functions then $N$ is a solid subset of ${\mathcal {L}}^{p}(\mu )$ that is not a band.

## Properties

The intersection of an arbitrary family of bands in a vector lattice $X$ is a band in $X.$

