Regularity for relational algebras and approach spacesRegularity for relational algebras and approach spaces
Faculty of Sciences. Mathematics and Computer Science
Topology and its applications. - Amsterdam
200(2016), p. 79-100
University of Antwerp
In this paper we consider relational T-algebras, objects in (T, 2)-Cat, as spaces and we explore the topological property of T-regularity. This notion goes back to Mobus  who introduced it in a more general abstract framework. When applied to the ultrafilter monad beta and to the well known lax-algebraic presentation of Top as (beta, 2)-Cat, beta 3-regularity is known to be equivalent to the usual regularity of the topological space . 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 (0, 2)-Cat, via the power-enriched functional ideal monad 0, 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 II with the initial extension to ReI, 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. (C) 2015 Elsevier B.V. All rights reserved.