Title 



Regularity for relational algebras and approach spaces
 
Author 



 
Abstract 



In this paper we consider relational TTalgebras, objects in (T,2)(T,2)CatCat, as spaces and we explore the topological property of TTregularity. This notion goes back to Möbus [18] who introduced it in a more general abstract framework. When applied to the ultrafilter monad Fullsize image (<1 K) and to the well known laxalgebraic presentation of TopTop as (Fullsize image (<1 K),2)(,2)CatCat, Fullsize image (<1 K)regularity is known to be equivalent to the usual regularity of the topological space [5]. We prove that in general for a powerenriched monad TT with the Kleisli extension, even when restricting to proper elements, TTregularity is too strong since in most cases it implies the object being indiscrete. For the laxalgebraic presentations of TopTop as (F,2)(F,2)CatCat, via the powerenriched filter monad FF and of AppApp as (I,2)(I,2)CatCat, via the powerenriched 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 laxalgebraic 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 BBregularity (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.   
Language 



English
 
Source (journal) 



Topology and its applications.  Amsterdam  
Publication 



Amsterdam : 2015
 
ISSN 



01668641
 
Volume/pages 



(2015), p. 122
 
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