Band (order theory) explained

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

l{L} ^{p(\mu)} (1 \leq p \leq inf ty ) is the usual space of real valued functions used to define Lp spaces L ^{p,} then l{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 l{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.