A Stone-type duality for stratified Alexandrov L-topological spaces

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• 作者：Wei Yao^a ; ^{yaowei0516@163.com" class="auth_mail" title="E-mail the corresponding author} ; Sang-Eon Han^b ; ^{sehan@jbnu.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Topology ; Category theory ; Duality ; Complete (co)prime element ; Complete lattice homomorphism ; Completely generated lattice ; (Complete) L-ordered set ; Alexandrov L-topology ; sT0sT0 space
• 刊名：Fuzzy Sets and Systems
• 出版年：2016
• 出版时间：1 January 2016
• 年：2016
• 卷：282
• 期：Complete
• 页码：1-20
• 全文大小：442 K

文摘

We show that for every complete lattice A  , both the set of completely prime elements and the set of completely coprime elements are one-to-one corresponding to class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si2.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=a90dc32ea3324cfc5c1b1bfa84053bea" title="Click to view the MathML source">CH(A)class="mathContainer hidden">class="mathCode">$\mathrm{CH}(A)$, the set of complete lattice homomorphisms from A to the two-element lattice class="boldFont">2. A complete lattice is called completely generated, a cg-lattice for short, if it is generated by the set of completely prime elements, or equivalently, by the set of completely coprime elements. Then we restudy the duality between the category of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si271.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=9319b33b61f00aea7f4664cf07a03c6c" title="Click to view the MathML source">T₀class="mathContainer hidden">class="mathCode">$T_0$ Alexandrov topological spaces and the category of cg-lattices by means of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si2.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=a90dc32ea3324cfc5c1b1bfa84053bea" title="Click to view the MathML source">CH(A)class="mathContainer hidden">class="mathCode">$\mathrm{CH}(A)$. With these preparations, for a frame L   as the truth value table, we introduce class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si1.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=9190c4420f08332b36dd4853405481ab" title="Click to view the MathML source">sT₀class="mathContainer hidden">class="mathCode">$s{T}_0$ separation axiom for stratified Alexandrov L  -topological spaces, and finally establish a duality between the category of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si1.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=9190c4420f08332b36dd4853405481ab" title="Click to view the MathML source">sT₀class="mathContainer hidden">class="mathCode">$s{T}_0$ stratified Alexandrov L-topological spaces and the category of completely generated complete L  -ordered sets. We also investigate some properties of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0165011414005843&_mathId=si1.gif&_user=111111111&_pii=S0165011414005843&_rdoc=1&_issn=01650114&md5=9190c4420f08332b36dd4853405481ab" title="Click to view the MathML source">sT₀class="mathContainer hidden">class="mathCode">$s{T}_0$ axiom, for example, it is hereditary by closed subspaces and productive.