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 lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si1.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=cc94a3017d5e610722c9a6b7426a2480" title="Click to view the MathML source">Tlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">T-algebras, objects in lsi2" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si2.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=395114e9f521dc44691cdd30a3fa71a9" title="Click to view the MathML source">(T,2)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(le-struck">T,2lse">)-lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si3.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=335fff4ec46bf5685004552e166c200d" title="Click to view the MathML source">Catlass="mathContainer hidden">lass="mathCode">low="scroll">Cat, as spaces and we explore the topological property of lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si1.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=cc94a3017d5e610722c9a6b7426a2480" title="Click to view the MathML source">Tlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">T-regularity. This notion goes back to M&ouml;bus [18] who introduced it in a more general abstract framework. When applied to the ultrafilter monad lass="imgLazyJSB inlineImage" height="14" width="7" alt="Full-size image (<1 K)" title="Full-size image (<1 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166864115005453-fx001.gif"> and to the well known lax-algebraic presentation of lsi23" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si23.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=ee6c586584ab056ff281dcb1a1b0f69f" title="Click to view the MathML source">Toplass="mathContainer hidden">lass="mathCode">low="scroll">Top as lsi5" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si5.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=4b9219c744e0552c039db3b973b4d3ae" title="Click to view the MathML source">(lass="imgLazyJSB inlineImage" height="14" width="7" alt="Full-size image (<1 K)" title="Full-size image (<1 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166864115005453-fx001.gif">,2)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(line-figure baseline="0.0"><link locator="fx001" type="simple" href="pii:S0166-8641(15)00545-3/fx001">line-figure>,2lse">)-lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si3.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=335fff4ec46bf5685004552e166c200d" title="Click to view the MathML source">Catlass="mathContainer hidden">lass="mathCode">low="scroll">Cat, lass="imgLazyJSB inlineImage" height="14" width="7" alt="Full-size image (<1 K)" title="Full-size image (<1 K)" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166864115005453-fx001.gif">-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 lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si1.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=cc94a3017d5e610722c9a6b7426a2480" title="Click to view the MathML source">Tlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">T with the Kleisli extension, even when restricting to proper elements, lsi1" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si1.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=cc94a3017d5e610722c9a6b7426a2480" title="Click to view the MathML source">Tlass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">T-regularity is too strong since in most cases it implies the object being indiscrete.

For the lax-algebraic presentations of lsi23" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si23.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=ee6c586584ab056ff281dcb1a1b0f69f" title="Click to view the MathML source">Toplass="mathContainer hidden">lass="mathCode">low="scroll">Top as lsi6" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si6.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=0c872809ae07878c6a7be5b02d4a2554" title="Click to view the MathML source">(F,2)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(le-struck">F,2lse">)-lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si3.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=335fff4ec46bf5685004552e166c200d" title="Click to view the MathML source">Catlass="mathContainer hidden">lass="mathCode">low="scroll">Cat, via the power-enriched filter monad lsi7" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si7.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=155a0608b2eb28d678e7e6f9a2328ff3" title="Click to view the MathML source">Flass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">F and of lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si8.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=53bb2ae95c8f488c584c51bb2f7a660d" title="Click to view the MathML source">Applass="mathContainer hidden">lass="mathCode">low="scroll">App as lsi9" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si9.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=266e586143928971cfd582859238579f" title="Click to view the MathML source">(I,2)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(le-struck">I,2lse">)-lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si3.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=335fff4ec46bf5685004552e166c200d" title="Click to view the MathML source">Catlass="mathContainer hidden">lass="mathCode">low="scroll">Cat, via the power-enriched functional ideal monad lsi10" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si10.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=09846ec0e6eef350366269449958d675" title="Click to view the MathML source">Ilass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">I, we present weaker conditions in terms of convergence of filters and functional ideals respectively, equivalent to the usual regularity in lsi23" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si23.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=ee6c586584ab056ff281dcb1a1b0f69f" title="Click to view the MathML source">Toplass="mathContainer hidden">lass="mathCode">low="scroll">Top and lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si8.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=53bb2ae95c8f488c584c51bb2f7a660d" title="Click to view the MathML source">Applass="mathContainer hidden">lass="mathCode">low="scroll">App.

For the lax-algebraic presentation of lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si8.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=53bb2ae95c8f488c584c51bb2f7a660d" title="Click to view the MathML source">Applass="mathContainer hidden">lass="mathCode">low="scroll">App as lsi11" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si11.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=9b69168509369829a20a1fde3dabe66b" title="Click to view the MathML source">(B,2)lass="mathContainer hidden">lass="mathCode">low="scroll">lse">(le-struck">B,2lse">)-lsi3" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si3.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=335fff4ec46bf5685004552e166c200d" title="Click to view the MathML source">Catlass="mathContainer hidden">lass="mathCode">low="scroll">Cat, via the prime functional ideal monad lsi12" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si12.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=1930d9ec93ba48c955fd7cd73bdea7da" title="Click to view the MathML source">Blass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">B, a submonad of lsi10" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si10.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=09846ec0e6eef350366269449958d675" title="Click to view the MathML source">Ilass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">I with the initial extension to lsi101" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si101.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=4be3689f596cee56e4faee0ad1b166af" title="Click to view the MathML source">Rellass="mathContainer hidden">lass="mathCode">low="scroll">Rel, restricting to proper elements already gives more interesting results. We prove that lsi12" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si12.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=1930d9ec93ba48c955fd7cd73bdea7da" title="Click to view the MathML source">Blass="mathContainer hidden">lass="mathCode">low="scroll">le-struck">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 lsi8" class="mathmlsrc">lass="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166864115005453&_mathId=si8.gif&_user=111111111&_pii=S0166864115005453&_rdoc=1&_issn=01668641&md5=53bb2ae95c8f488c584c51bb2f7a660d" title="Click to view the MathML source">Applass="mathContainer hidden">lass="mathCode">low="scroll">App in terms of convergence of prime functional ideals.