Regularity for relational algebras and approach spaces

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• 作者：E. Colebunders^a ; ^{evacoleb@vub.ac.be" class="auth_mail" title="E-mail the corresponding author} ; R. Lowen^b ; ^{bob.lowen@uantwerpen.be" class="auth_mail" title="E-mail the corresponding author} ; K. Van Opdenbosch^a ; ^{karen.van.opdenbosch@vub.ac.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：18B30 ; 18C20 ; 54A05 ; 54A20 ; 54B30 ; 54E99
• 刊名：Topology and its Applications
• 出版年：2016
• 出版时间：1 March 2016
• 年：2016
• 卷：200
• 期：Complete
• 页码：79-100
• 全文大小：477 K

文摘

In this paper we consider relational T$\mathbb{T}$-algebras, objects in (T,2)$(\mathbb{T},2)$-c200d" title="Click to view the MathML source">Cat$\mathsf{Cat}$, as spaces and we explore the topological property of T$\mathbb{T}$-regularity. This notion goes back to Möbus [18] who introduced it in a more general abstract framework. When applied to the ultrafilter monad and to the well known lax-algebraic presentation of Top$\mathsf{Top}$ as (,2)$(\text{<inline-figure baseline="0.0"><link locator="fx001" type="simple" href="pii:S0166-8641(15)00545-3/fx001"></inline-figure>},2)$-c200d" title="Click to view the MathML source">Cat$\mathsf{Cat}$, -regularity is known to be equivalent to the usual regularity of the topological space [5]. We prove that in general for a power-enriched monad T$\mathbb{T}$ with the Kleisli extension, even when restricting to proper elements, T$\mathbb{T}$-regularity is too strong since in most cases it implies the object being indiscrete.

For the lax-algebraic presentations of Top$\mathsf{Top}$ as (F,2)$(\mathbb{F},2)$-c200d" title="Click to view the MathML source">Cat$\mathsf{Cat}$, via the power-enriched filter monad F$\mathbb{F}$ and of App$\mathsf{App}$ as (I,2)$(\mathbb{I},2)$-c200d" title="Click to view the MathML source">Cat$\mathsf{Cat}$, via the power-enriched functional ideal monad I$\mathbb{I}$, we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in Top$\mathsf{Top}$ and App$\mathsf{App}$.

For the lax-algebraic presentation of App$\mathsf{App}$ as (B,2)$(\mathbb{B},2)$-c200d" title="Click to view the MathML source">Cat$\mathsf{Cat}$, via the prime functional ideal monad B$\mathbb{B}$, a submonad of I$\mathbb{I}$ with the initial extension to Rel$\mathsf{Rel}$, restricting to proper elements already gives more interesting results. We prove that B$\mathbb{B}$-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$\mathsf{App}$ in terms of convergence of prime functional ideals.