Regularity for relational algebras and approach spaces
Faculty of Sciences. Mathematics and Computer Science
Topology and its applications. - Amsterdam
, p. 1-22
University of Antwerp
In this paper we consider relational TT-algebras, objects in (T,2)(T,2)-CatCat, as spaces and we explore the topological property of TT-regularity. This notion goes back to Möbus  who introduced it in a more general abstract framework. When applied to the ultrafilter monad Full-size image (<1 K) and to the well known lax-algebraic presentation of TopTop as (Full-size image (<1 K),2)(,2)-CatCat, Full-size image (<1 K)-regularity is known to be equivalent to the usual regularity of the topological space . We prove that in general for a power-enriched monad TT with the Kleisli extension, even when restricting to proper elements, TT-regularity is too strong since in most cases it implies the object being indiscrete. For the lax-algebraic presentations of TopTop as (F,2)(F,2)-CatCat, via the power-enriched filter monad FF and of AppApp as (I,2)(I,2)-CatCat, via the power-enriched functional ideal monad II, we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in TopTop and AppApp. For the lax-algebraic presentation of AppApp as (B,2)(B,2)-CatCat, via the prime functional ideal monad BB, a submonad of II with the initial extension to RelRel, restricting to proper elements already gives more interesting results. We prove that BB-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 AppApp in terms of convergence of prime functional ideals.