{"title":"Quantum B-modules","authors":"Xia Zhang, Wolfgang Rump","doi":"10.1002/malq.202100029","DOIUrl":null,"url":null,"abstract":"<p>Quantum B-algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non-commutative arithmetic, and quantum theory. A quantum B-algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B-module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi-poset version of the completion of Dedekind and MacNeille.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2022-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Logic Quarterly","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/malq.202100029","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"LOGIC","Score":null,"Total":0}
引用次数: 1
Abstract
Quantum B-algebras are partially ordered algebras characterizing the residuated structure of a quantale. Examples arise in algebraic logic, non-commutative arithmetic, and quantum theory. A quantum B-algebra with trivial partial order is equivalent to a group. The paper introduces a corresponding analogue of quantale modules. It is proved that every quantum B-module admits an injective envelope which is a quantale module. The injective envelope is constructed explicitly as a completion, a multi-poset version of the completion of Dedekind and MacNeille.
期刊介绍：
Mathematical Logic Quarterly publishes original contributions on mathematical logic and foundations of mathematics and related areas, such as general logic, model theory, recursion theory, set theory, proof theory and constructive mathematics, algebraic logic, nonstandard models, and logical aspects of theoretical computer science.
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