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 a band.

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