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"> 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">-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">, 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"> 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"> 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">-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">, 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">, 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"> 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">.
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"> 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">-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">, 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">, 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"> 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">, 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">-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"> in terms of convergence of prime functional ideals.