Kleisli monoids describing approach spaces

We study the functional ideal monad I = (I, m, e)$ on Set and show that this monad is power-enriched. This leads us to the category I-Mon of all I-monoids with structure preserving maps. We show that this category is isomorphic to App, the category of approach spaces with contractions as morphisms. Through the concrete isomorphism, an I-monoid (X,u) corresponds to an approach space (X, A) described in terms of its bounded local approach system. When I is extended to Rel$ using the Kleisli extension, from on the fact that I-Mon and (I,2)-Cat are isomorphic, we obtain the result that App can be isomorphically described in terms of convergence of functional ideals, based on the two axioms of relational algebras, reflexivity and transitivity.
We compare these axioms to the ones put forward in Index Analysis.
Considering the submonad B of all prime functional ideals, we show that it is both sup-dense and interpolating in I, from which we get that (B,2)-Cat and (I,2)-Cat are isomorphic. We present some simple axioms describing App in terms of prime functional ideal convergence.