{"title":"Rim-Perfect Compactifications of Frames and Zero-Dimensionally Embedded Remainders","authors":"Simo Mthethwa, Gugulethu Nogwebela","doi":"10.1007/s41980-024-00896-7","DOIUrl":null,"url":null,"abstract":"<p>We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales <i>S</i> with the property that the ambient frame has a basis, elements of which induce open sublocales whose frontiers miss <i>S</i>. This notion is stronger than the traditional zero-dimensionality of a sublocale. A compactification of a frame is perfect if its associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect compactifications of frames is introduced, and we show that it contains all the perfect ones. Indeed, not every rim-perfect compactification is perfect, but compactifications with a ZDE remainder do not distinguish between rim-perfectness and perfectness. The Freudenthal compactification has a ZDE remainder. We show that a frame <i>L</i> is rim-compact if and only if <i>L</i> has a compactification with a ZDE remainder. Several results concerning perfect compactifications and ZDE remainders are provided.</p>","PeriodicalId":9395,"journal":{"name":"Bulletin of The Iranian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of The Iranian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s41980-024-00896-7","RegionNum":4,"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
We introduce zero-dimensionally embedded (ZDE) sublocales as those sublocales S with the property that the ambient frame has a basis, elements of which induce open sublocales whose frontiers miss S. This notion is stronger than the traditional zero-dimensionality of a sublocale. A compactification of a frame is perfect if its associated right adjoint preserves disjoint binary joins. Herein, the class of rim-perfect compactifications of frames is introduced, and we show that it contains all the perfect ones. Indeed, not every rim-perfect compactification is perfect, but compactifications with a ZDE remainder do not distinguish between rim-perfectness and perfectness. The Freudenthal compactification has a ZDE remainder. We show that a frame L is rim-compact if and only if L has a compactification with a ZDE remainder. Several results concerning perfect compactifications and ZDE remainders are provided.
我们将零维度嵌入（ZDE）子域引入那些具有环境框架有一个基的属性的子域 S，这些基的元素会诱导出边界错过 S 的开放子域。如果一个框架的相关右邻接保留了不相交的二元连接，那么这个框架的紧凑化就是完美的。在这里，我们引入了框架的边缘完美压缩类，并证明它包含了所有的完美压缩。事实上，并不是每一个边缘完美紧凑都是完美的，但是有 ZDE 余数的紧凑并不区分边缘完美和完美。弗赖登塔尔紧缩具有 ZDE 余数。我们证明，当且仅当 L 具有 ZDE 余数的紧凑化时，框架 L 才是边缘紧凑的。我们还提供了一些关于完美紧凑性和 ZDE 余数的结果。
期刊介绍：
The Bulletin of the Iranian Mathematical Society (BIMS) publishes original research papers as well as survey articles on a variety of hot topics from distinguished mathematicians. Research papers presented comprise of innovative contributions while expository survey articles feature important results that appeal to a broad audience. Articles are expected to address active research topics and are required to cite existing (including recent) relevant literature appropriately. Papers are critically reviewed on the basis of quality in its exposition, brevity, potential applications, motivation, value and originality of the results. The BIMS takes a high standard policy against any type plagiarism. The editorial board is devoted to solicit expert referees for a fast and unbiased review process.
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