  We show that for an ideal $H$ in an Archimedean vector lattice $F$ the
following conditions are equivalent:
  $\bullet$ $H$ is a projection band;
  $\bullet$ Any collection of mutually disjoint vectors in $H$, which is order
bounded in $F$, is order bounded in $H$;
  $\bullet$ $H$ is an infinite meet-distributive element of the lattice
$\mathcal{I}_{F}$ of all ideals in $F$ in the sense that $\bigcap\limits_{J\in
\mathcal{J}}\left(H+ J\right)=H+ \bigcap \mathcal{J}$, for any
$\mathcal{J}\subset \mathcal{I}_{F}$.
  Additionally, we show that if $F$ is uniformly complete and $H$ is a
uniformly closed principal ideal, then $H$ is a projection band. In the process
we investigate some order properties of lattices of continuous functions on
Tychonoff topological spaces.

Functional Analysis;General Topology