  We study a duality for the $n$-point functions in VEV formalism that we call
the ordinary vs fully simple duality. It provides an ultimate generalisation
and a proper context for the duality between maps and fully simple maps
observed by Borot and Garcia-Failde. Our approach allows to transfer the
algebraicity properties between the systems of $n$-point functions related by
this duality, and gives direct tools for the analysis of singularities. As an
application, we give a proof of a recent conjecture of Borot and Garcia-Failde
on topological recursion for fully simple maps.
