Regularity for relational algebras and approach spaces

In this paper we consider relational T-algebras, objects in (T,2)-Cat; as spaces
and we explore the topological property of T-regularity. We prove that in general for a power-enriched monad T with the Kleisli extension, even when restricting to proper elements, T-regularity is too strong since in most cases it implies the object being indiscrete. For the lax-algebraic presentations of Top as (F,2)-Cat; via the power-enriched filter monad F and of App as (I,2)-Cat, via the power-enriched functional ideal monad I we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in Top and App.
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 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.