{"title":"Topological representation of some lattices","authors":"Ali Taherifar, Mohamad Reza Ahmadi Zand","doi":"10.1515/ms-2024-0021","DOIUrl":null,"url":null,"abstract":"In this paper, we first give a topological representation of some algebraic lattices of ideals of <jats:italic>C</jats:italic>(<jats:italic>X</jats:italic>). Next, we apply these results and prove that a space <jats:italic>X</jats:italic> is normal if and only if the lattice of closed fixed ideals of <jats:italic>C</jats:italic>(<jats:italic>X</jats:italic>) is a sublattice of the lattice of ideals of <jats:italic>C</jats:italic>(<jats:italic>X</jats:italic>). It is proved that if two rings <jats:italic>C</jats:italic>(<jats:italic>X</jats:italic>) and <jats:italic>C</jats:italic>(<jats:italic>Y</jats:italic>) are isomorphic, then two lattices <jats:italic>Z</jats:italic> <jats:sup>∘</jats:sup>[<jats:italic>X</jats:italic>] and <jats:italic>Z</jats:italic> <jats:sup>∘</jats:sup>[<jats:italic>Y</jats:italic>] are isomorphic. We conclude that two rings <jats:italic>C</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>X</jats:italic>) and <jats:italic>C</jats:italic> <jats:sup>*</jats:sup>(<jats:italic>Y</jats:italic>) are isomorphic if and only if two lattices <jats:italic>Z</jats:italic>[<jats:italic>βX</jats:italic>] and <jats:italic>Z</jats:italic>[<jats:italic>βY</jats:italic>] are isomorphic.","PeriodicalId":18282,"journal":{"name":"Mathematica Slovaca","volume":"10 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematica Slovaca","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1515/ms-2024-0021","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we first give a topological representation of some algebraic lattices of ideals of C(X). Next, we apply these results and prove that a space X is normal if and only if the lattice of closed fixed ideals of C(X) is a sublattice of the lattice of ideals of C(X). It is proved that if two rings C(X) and C(Y) are isomorphic, then two lattices Z∘[X] and Z∘[Y] are isomorphic. We conclude that two rings C*(X) and C*(Y) are isomorphic if and only if two lattices Z[βX] and Z[βY] are isomorphic.
期刊介绍：
Mathematica Slovaca, the oldest and best mathematical journal in Slovakia, was founded in 1951 at the Mathematical Institute of the Slovak Academy of Science, Bratislava. It covers practically all mathematical areas. As a respectful international mathematical journal, it publishes only highly nontrivial original articles with complete proofs by assuring a high quality reviewing process. Its reputation was approved by many outstanding mathematicians who already contributed to Math. Slovaca. It makes bridges among mathematics, physics, soft computing, cryptography, biology, economy, measuring, etc. The Journal publishes original articles with complete proofs. Besides short notes the journal publishes also surveys as well as some issues are focusing on a theme of current interest.
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