Proposition 4

For an ordered group $G$ to be filtered it is necessary and sufficient that it be generated by its positive elements, that is that every element be the difference of two positive elements.

Indeed if $G$ is filtered, then for each $x \in G$ there exists a positive element $z$ such that $x \leq z$, and $x$ is the difference of the two positive elements $z$ and $z - x$. Conversely, if $x = u - v$ and $y = w - t$ with $u, v, w, t$ positive, then the element $u + w$ is greater than $x$ and greater than $y$.

1 The terminology used in loc. cit. is directed.