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Algebra Universalis最新文献

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Catoids and modal convolution algebras Catoids与模态卷积代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-02-25 DOI: 10.1007/s00012-023-00805-9
Uli Fahrenberg, Christian Johansen, Georg Struth, Krzysztof Ziemiański
{"title":"Catoids and modal convolution algebras","authors":"Uli Fahrenberg,&nbsp;Christian Johansen,&nbsp;Georg Struth,&nbsp;Krzysztof Ziemiański","doi":"10.1007/s00012-023-00805-9","DOIUrl":"10.1007/s00012-023-00805-9","url":null,"abstract":"<div><p>We show how modal quantales arise as convolution algebras <span>(Q^X)</span> of functions from catoids <i>X</i>, multisemigroups equipped with source and target maps, into modal quantales value or weight quantales <i>Q</i>. In the tradition of boolean algebras with operators we study modal correspondences between algebraic laws in <i>X</i>, <i>Q</i> and <span>(Q^X)</span>. The catoids introduced generalise Schweizer and Sklar’s function systems and single-set categories to structures isomorphic to algebras of ternary relations, as they are used for boolean algebras with operators and substructural logics. Our correspondence results support a generic construction of weighted modal quantales from catoids. This construction is illustrated by many examples. We also relate our results to reasoning with stochastic matrices or probabilistic predicate transformers.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00805-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45736966","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
Varieties of ordered algebras as categories 作为范畴的有序代数的变种
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-02-24 DOI: 10.1007/s00012-023-00806-8
Jiří Adámek, Jiří Rosický
{"title":"Varieties of ordered algebras as categories","authors":"Jiří Adámek,&nbsp;Jiří Rosický","doi":"10.1007/s00012-023-00806-8","DOIUrl":"10.1007/s00012-023-00806-8","url":null,"abstract":"<div><p>A variety is a category of ordered (finitary) algebras presented by inequations between terms. We characterize categories enriched over the category of posets which are equivalent to a variety. This is quite analogous to Lawvere’s classical characterization of varieties of ordinary algebras. We also study the relationship of varieties to discrete Lawvere theories, and varieties as concrete categories over <span>(mathbf{ Pos })</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00806-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42967837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Adjunction of a strong unit to a hyper-archimedean lattice-ordered group 强单元对超阿基米德格序群的附加
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-02-13 DOI: 10.1007/s00012-023-00803-x
Philip Scowcroft
{"title":"Adjunction of a strong unit to a hyper-archimedean lattice-ordered group","authors":"Philip Scowcroft","doi":"10.1007/s00012-023-00803-x","DOIUrl":"10.1007/s00012-023-00803-x","url":null,"abstract":"<div><p>This paper studies conditions in which a hyperarchimedean lattice-ordered group embeds into a hyperarchimedean lattice-ordered group with strong unit. While Conrad and Martinez showed that some hyperarchimedean lattice-ordered groups do not admit such embeddings, Section 3 presents a sufficient condition, in terms of the generalized Boolean algebra of principal <span>(ell )</span>-ideals, for the existence of such an embedding. Section 4 presents new examples of hyperarchimedean lattice-ordered groups not admitting such embeddings, while Section 5 shows that even when such an embedding exists, adjunction of a strong unit may yield non-isomorphic hyperarchimedean extensions. Section 6 shows that if one assumes the existence of weakly compact cardinals, then the sufficient condition from Section 3 is not necessary; and Section 7 studies the logical complexity of the condition “embeddable into a hyperarchimedean lattice-ordered group with strong unit.”</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00803-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48022327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Lacunary series, algebraic normal forms, convolutions 缺元级数，代数范式，卷积
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-02-11 DOI: 10.1007/s00012-023-00808-6
Arthur Knoebel
{"title":"Lacunary series, algebraic normal forms, convolutions","authors":"Arthur Knoebel","doi":"10.1007/s00012-023-00808-6","DOIUrl":"10.1007/s00012-023-00808-6","url":null,"abstract":"<div><p>Conditions are found on an operation on a finite set for its transform to be lacunary, that is, missing many expected terms. Often the condition is that the operation preserves a relation. General operations are split into odd and even lacunary parts, or more generally, into several lacunary parts given by a non-binary parity. This is applied to classical Fourier transforms as well as algebraic normal forms. With this theory, an explicit polynomial expansion is given for any operation in a preprimal algebra based on a finite elementary Abelian group. Convolution is defined, with a criterion given for it to commute with transforms.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45774288","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Duality for normal lattice expansions and sorted residuated frames with relations 正规格展开式的对偶性和带关系的排序残馀框架
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-01-29 DOI: 10.1007/s00012-023-00802-y
Chrysafis Hartonas
{"title":"Duality for normal lattice expansions and sorted residuated frames with relations","authors":"Chrysafis Hartonas","doi":"10.1007/s00012-023-00802-y","DOIUrl":"10.1007/s00012-023-00802-y","url":null,"abstract":"<div><p>We revisit the problem of Stone duality for lattices with quasioperators, presenting a fresh duality result. The new result is an improvement over that of our previous work in two important respects. First, the axiomatization of frames is now simplified, partly by incorporating Gehrke’s proposal of section stability for relations. Second, morphisms are redefined so as to preserve Galois stable (and co-stable) sets and we rely for this, partly again, on Goldblatt’s recently proposed definition of bounded morphisms for polarities. In studying the dual algebraic structures associated to polarities with relations we demonstrate that stable/co-stable set operators result as the Galois closure of the restriction of classical (though sorted) image operators generated by the frame relations to Galois stable/co-stable sets. This provides a proof, at the representation level, that non-distributive logics can be regarded as fragments of sorted residuated (poly)modal logics, a research direction recently initiated by this author.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00802-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45020831","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
What makes a Stone topological algebra Profinite 是什么让石头拓扑代数成为无限的
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-01-29 DOI: 10.1007/s00012-023-00804-w
Jorge Almeida, Herman Goulet-Ouellet, Ondřej Klíma
{"title":"What makes a Stone topological algebra Profinite","authors":"Jorge Almeida,&nbsp;Herman Goulet-Ouellet,&nbsp;Ondřej Klíma","doi":"10.1007/s00012-023-00804-w","DOIUrl":"10.1007/s00012-023-00804-w","url":null,"abstract":"<div><p>This paper is a contribution to understanding what properties should a topological algebra on a Stone space satisfy to be profinite. We reformulate and simplify proofs for some known properties using syntactic congruences. We also clarify the role of various alternative ways of describing syntactic congruences, namely by finite sets of terms and by compact sets of continuous self mappings of the algebra.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00804-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48876626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Connected topological lattice-ordered groups 连通拓扑格序群
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-01-10 DOI: 10.1007/s00012-022-00800-6
Francis Jordan
{"title":"Connected topological lattice-ordered groups","authors":"Francis Jordan","doi":"10.1007/s00012-022-00800-6","DOIUrl":"10.1007/s00012-022-00800-6","url":null,"abstract":"<div><p>We answer two open problems about lattice-ordered groups that admit a connected lattice-ordered group topology. We show that, in the general case, admitting a connected lattice-ordered group topology does not effect the algebraic structure of the lattice-ordered group. For example, admitting a connected lattice-ordered group topology does not imply that the lattice-ordered group is Archimedean or even representable. On the other hand, if one assumes that the lattice-ordered group has a basis, then admitting a lattice-ordered group topology implies that the lattice-ordered group is a subdirect product of copies of the real numbers.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42319879","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Algebraic frames in which dense elements are above dense compact elements 稠密元素在稠密紧元素之上的代数框架
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-12-03 DOI: 10.1007/s00012-022-00799-w
Themba Dube, Siphamandla Blose
{"title":"Algebraic frames in which dense elements are above dense compact elements","authors":"Themba Dube,&nbsp;Siphamandla Blose","doi":"10.1007/s00012-022-00799-w","DOIUrl":"10.1007/s00012-022-00799-w","url":null,"abstract":"<div><p>A ring is called a zip ring (Carl Faith coined this term) if every faithful ideal contains a finitely generated faithful ideal. By first proving that a reduced ring is a zip ring if and only if every dense element of the frame of its radical ideals is above a compact dense element, we study algebraic frames with the property stated in the title. We call them zipped. They generalize the coherent frames of radical ideals of zip rings, but (unlike coherent frames) they need not be compact. The class of zipped algebraic frames is closed under finite products, but not under infinite products. If the coproduct of two algebraic frames is zipped, then each cofactor is zipped. If the ring is not necessarily reduced, then its frame of radical ideals is zipped precisely when the ring satisfies what in the literature is called the weak zip property. For a Tychonoff space <i>X</i>, we show that <i>C</i>(<i>X</i>) is a zip ring if and only if <i>X</i> is a finite discrete space.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49612562","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Projectivity in (bounded) commutative integral residuated lattices （有界）交换积分剩余格的射影性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-11-29 DOI: 10.1007/s00012-022-00798-x
Paolo Aglianò, Sara Ugolini
{"title":"Projectivity in (bounded) commutative integral residuated lattices","authors":"Paolo Aglianò,&nbsp;Sara Ugolini","doi":"10.1007/s00012-022-00798-x","DOIUrl":"10.1007/s00012-022-00798-x","url":null,"abstract":"<div><p>In this paper, we study projective algebras in varieties of (bounded) commutative integral residuated lattices. We make use of a well-established construction in residuated lattices, the ordinal sum, and the order property of divisibility. Via the connection between projective and splitting algebras, we show that the only finite projective algebra in <span>(mathsf {{FL}_{ew}})</span> is the two-element Boolean algebra. Moreover, we show that several interesting varieties have the property that every finitely presented algebra is projective, such as locally finite varieties of hoops. Furthermore, we show characterization results for finite projective Heyting algebras, and finitely generated projective algebras in locally finite varieties of bounded hoops and BL-algebras. Finally, we connect our results with the algebraic theory of unification.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45112825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 5
On the number of universal algebraic geometries 关于泛代数几何的个数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-11-28 DOI: 10.1007/s00012-022-00797-y
Erhard Aichinger, Bernardo Rossi
{"title":"On the number of universal algebraic geometries","authors":"Erhard Aichinger,&nbsp;Bernardo Rossi","doi":"10.1007/s00012-022-00797-y","DOIUrl":"10.1007/s00012-022-00797-y","url":null,"abstract":"<div><p>The <i>algebraic geometry</i> of a universal algebra <span>({textbf{A}})</span> is defined as the collection of solution sets of systems of term equations. Two algebras <span>({textbf{A}}_1)</span> and <span>({textbf{A}}_2)</span> are called <i>algebraically equivalent</i> if they have the same algebraic geometry. We prove that on a finite set <i>A</i> with <span>(|A|)</span> there are countably many algebraically inequivalent Mal’cev algebras and that on a finite set <i>A</i> with <span>(|A|)</span> there are continuously many algebraically inequivalent algebras.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-022-00797-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45043025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
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