  Disjointness, bands, and band projections are a classical and essential part
of the structure theory of vector lattices. If $X$ is such a lattice, those
notions seem - at first glance - intimately related to the lattice operations
on $X$. The last fifteen years, though, have seen an extension of all those
concepts to a much larger class of ordered vector spaces.
  In fact if $X$ is an Archimedean ordered vector space with generating cone,
or a member of the slightly larger class of pre-Riesz spaces, then the notions
of disjointness, bands and band projections can be given proper meaning and
give rise to a non-trivial structure theory.
  The purpose of this note is twofold: (i) We show that, on any pre-Riesz
space, the structure of the space of all band projections is remarkably close
to what we have in the case of vector lattices. In particular, this space is a
Boolean algebra. (ii) We give several criteria for a pre-Riesz space to already
be a vector lattice. These criteria are coined in terms of disjointness and
closely related concepts, and they mark how lattice-like the order structure of
pre-Riesz spaces can get before the theory collapses to the vector lattice
case.
