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 ${\displaystyle x\in M.}$ The smallest band containing a subset ${\displaystyle S}$ of ${\displaystyle X}$ is called the band generated by ${\displaystyle S}$ in ${\displaystyle X.}$ A band generated by a singleton set is called a principal band.