{"title":"Semilinear De Morgan monoids and epimorphisms","authors":"J. J. Wannenburg, J. G. Raftery","doi":"10.1007/s00012-023-00837-1","DOIUrl":null,"url":null,"abstract":"<div><p>A representation theorem is proved for De Morgan monoids that are (i) <i>semilinear</i>, i.e., subdirect products of totally ordered algebras, and (ii) <i>negatively generated</i>, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00837-1.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algebra Universalis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00012-023-00837-1","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A representation theorem is proved for De Morgan monoids that are (i) semilinear, i.e., subdirect products of totally ordered algebras, and (ii) negatively generated, i.e., generated by lower bounds of the neutral element. Using this theorem, we prove that the De Morgan monoids satisfying (i) and (ii) form a variety—in fact, a locally finite variety. We then prove that epimorphisms are surjective in every variety of negatively generated semilinear De Morgan monoids. In the process, epimorphism-surjectivity is established for several other classes as well, including the variety of all semilinear idempotent commutative residuated lattices and all varieties of negatively generated semilinear Dunn monoids. The results settle natural questions about Beth-style definability for a range of substructural logics.
我们证明了德摩根单元的表示定理：(i) 半线性，即完全有序代数的子直积；(ii) 负生成，即由中性元素的下界生成。利用这个定理，我们证明了满足 (i) 和 (ii) 的 De Morgan 单元构成了一个变式--事实上是一个局部有限变式。然后，我们证明在负生成的半线性 De Morgan 单元的每一个变种中，外形变都是可射的。在此过程中，我们还建立了其他几类的外形射性，包括所有半线性幂交换残差格的变种和所有负生成半线性邓恩单体的变种。这些结果解决了关于一系列子结构逻辑的贝思式可定义性的自然问题。
期刊介绍：
Algebra Universalis publishes papers in universal algebra, lattice theory, and related fields. In a pragmatic way, one could define the areas of interest of the journal as the union of the areas of interest of the members of the Editorial Board. In addition to research papers, we are also interested in publishing high quality survey articles.
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