Projection bands and atoms in pervasive pre-Riesz spaces

Abstract

In vector lattices, the concept of a projection band is a basic tool. We deal with projection bands in the more general setting of an Archimedean pre-Riesz space $X$. We relate them to projection bands in a vector lattice cover $Y$ of $X$. If $X$ is pervasive, then a projection band in $X$ extends to a projection band in $Y$, whereas the restriction of a projection band $B$ in $Y$ is not a projection band in $X$, in general. We give conditions under which the restriction of $B$ is a projection band in $X$. We introduce atoms and discrete elements in $X$ and show that every atom is discrete. The converse implication is true, provided $X$ is pervasive. In this setting, we link atoms in $X$ to atoms in $Y$. If $X$ contains an atom $a>0$, we show that the principal band generated by $a$ is a projection band. Using atoms in a finite dimensional Archimedean pre-Riesz space $X$, we establish that $X$ is pervasive if and only if it is a vector lattice.