Band (order theory)

In mathematics, specifically in order theory and functional analysis, a band in a vector lattice ${\displaystyle X}$ is a subspace ${\displaystyle M}$ of ${\displaystyle X}$ that is solid and such that for all ${\displaystyle S\subseteq M}$ such that ${\displaystyle x=\sup S}$ exists in ${\displaystyle X,}$ we have ${\displaystyle x\in M.}$^[1] The smallest band containing a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called the band generated by ${\displaystyle S}$ in ${\displaystyle X.}$^[1] A band generated by a singleton set is called a principal band.

Not to be confused with Band (algebra).

Examples

For any subset ${\displaystyle S}$ of a vector lattice ${\displaystyle X,}$ the set ${\displaystyle S^{\perp }}$ of all elements of ${\displaystyle X}$ disjoint from ${\displaystyle S}$ is a band in ${\displaystyle X.}$^[1]

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

Properties

The intersection of an arbitrary family of bands in a vector lattice ${\displaystyle X}$ is a band in ${\displaystyle X.}$^[2]

See also

• Solid set
• Locally convex vector lattice
• Vector lattice – Partially ordered vector space, ordered as a latticePages displaying short descriptions of redirect targets