Regularity for relational algebras and approach spaces

For the lax-algebraic presentation of App as (B,2)-Cat, via the prime functional ideal monad B, a submonad of I with the initial extension to Rel, restricting to proper elements already gives more interesting results. We prove that B-regularity (restricted to proper prime functional ideals) is equivalent to the approach space being topological and regular. However it requires further weakening of the concept to obtain a characterization of the usual regularity in App in terms of convergence of prime functional ideals.