Characterizations of the projection bands and some order properties of the lattices of continuous functions

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 $\bigcap _{J\in \mathcal{J}}(H+J)=H+\bigcap \mathcal{J}$, 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.