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 $S^$ of all elements of $X$ disjoint from $S$ is a band in $X.$

If $\mathcal^p(\mu)$ ($1 \leq p \leq \infty$) is the usual space of real valued functions used to define Lp spaces $L^p,$ then $\mathcal^p(\mu)$ 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 $\mathcal^p(\mu)$ that is a band.

Properties

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

See also

*
*
*

References

*
*

{{Functional analysis

Functional analysis