{"title":"Construction of Quantum B-algebras over Posets","authors":"Shengwei Han, Xin Wang, Congcong Wang","doi":"10.1007/s11083-024-09682-w","DOIUrl":null,"url":null,"abstract":"<p>In order to provide a unified semantics for non-commutative algebraic logic, based on posets, Rump and Yang introduced the concept of quantum <i>B</i>-algebras. In this paper, we mainly consider the construction of quantum <i>B</i>-algebras over posets. We prove that a finite poset can support a quantum <i>B</i>-algebra if and only if its every connected component has a greatest element. However, such a result for infinite posets is not necessarily true. Under certain conditions, some interesting results for a poset to support quantum <i>B</i>-algebra are provided.</p>","PeriodicalId":501237,"journal":{"name":"Order","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Order","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11083-024-09682-w","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
In order to provide a unified semantics for non-commutative algebraic logic, based on posets, Rump and Yang introduced the concept of quantum B-algebras. In this paper, we mainly consider the construction of quantum B-algebras over posets. We prove that a finite poset can support a quantum B-algebra if and only if its every connected component has a greatest element. However, such a result for infinite posets is not necessarily true. Under certain conditions, some interesting results for a poset to support quantum B-algebra are provided.
为了给基于正集的非交换代数逻辑提供统一的语义，Rump 和 Yang 引入了量子 B 带的概念。在本文中，我们主要考虑在正集上构造量子 B 带。我们证明，当且仅当一个有限正集的每个相连分量都有一个最大元素时，它可以支持一个量子 B-代数。然而，对于无限正集，这样的结果并不一定成立。在某些条件下，我们提供了正集支持量子 B 代数的一些有趣结果。
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