Addition of filters on the natural numbers ($A\in \mathcal{F}+\mathcal{G}⟺{\forall }_{\mathcal{F}}x{\forall }_{\mathcal{G}}y(x+y\in A)$)

For proper filters on positive integers, define $A\in \mathcal{F}+\mathcal{G}$ when $\{x:A-x\in \mathcal{G}\}\in \mathcal{F}$. This is again a proper filter. The inner truth-set map preserves the whole set, the empty set, inclusions and finite intersections; applying the outer filter gives all filter axioms. The order of the two filter quantifiers is part of the definition and is not silently interchangeable.