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Algebra Universalis最新文献

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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":"84 1","pages":""},"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":"84 1","pages":""},"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":"84 1","pages":""},"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":"84 1","pages":""},"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":"84 1","pages":""},"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":"84 1","pages":""},"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":"84 1","pages":""},"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
Another look on tense and related operators 时态及相关算子的再认识
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-11-01 DOI: 10.1007/s00012-022-00794-1
Michal Botur, Jan Paseka, Richard Smolka
{"title":"Another look on tense and related operators","authors":"Michal Botur,&nbsp;Jan Paseka,&nbsp;Richard Smolka","doi":"10.1007/s00012-022-00794-1","DOIUrl":"10.1007/s00012-022-00794-1","url":null,"abstract":"<div><p>Motivated by the classical work of Halmos on functional monadic Boolean algebras, we derive three basic sup-semilattice constructions, among other things, the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic, and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-022-00794-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46664865","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
The assembly of a pointfree bispace and its two variations 无点空间及其两个变体的组合
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-10-08 DOI: 10.1007/s00012-022-00793-2
Anna Laura Suarez
{"title":"The assembly of a pointfree bispace and its two variations","authors":"Anna Laura Suarez","doi":"10.1007/s00012-022-00793-2","DOIUrl":"10.1007/s00012-022-00793-2","url":null,"abstract":"<div><p>We explore a pointfree theory of bitopological spaces (that is, sets equipped with two topologies). In particular, here we regard <i>finitary biframes</i> as duals of bitopological spaces. In particular for a finitary biframe <span>(mathcal {L})</span> the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly of a finitary biframe <span>(mathcal {L})</span>, there are two other structures <span>(mathsf {A}_{cf}(mathcal {L}))</span> and <span>(mathsf {A}_{pm }(mathcal {L}))</span>, which both have the same main component as <span>(mathsf {A}(mathcal {L}))</span>. The main component of both <span>(mathsf {A}_{cf}(mathcal {L}))</span> and <span>(mathsf {A}_{pm }(mathcal {L}))</span> is the ordered collection of all biquotients of <span>(mathcal {L})</span>. The structure <span>(mathsf {A}_{cf}(mathcal {L}))</span> being a biframe shows that the collection of all biquotients is generated by the frame of the <i>patch-closed</i> biquotients together with that of the <i>patch-fitted</i> ones. The structure <span>(mathsf {A}_{pm }(mathcal {L}))</span> being a biframe shows the collection of all biquotients is generated by the frame of the <i>positive</i> biquotients together with that of the <i>negative</i> ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of two characterization theorems for these axioms hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise <span>(T_1)</span> are characterized, among other things in terms of the spectrum <span>(mathsf {bpt}(mathsf {A}_{cf}(mathcal {L})))</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-022-00793-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48148140","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
Uniform continuity of pointfree real functions via farness and related Galois connections 无点实函数的一致连续性及其相关Galois连接
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2022-10-03 DOI: 10.1007/s00012-022-00795-0
Ana Belén Avilez, Jorge Picado
{"title":"Uniform continuity of pointfree real functions via farness and related Galois connections","authors":"Ana Belén Avilez,&nbsp;Jorge Picado","doi":"10.1007/s00012-022-00795-0","DOIUrl":"10.1007/s00012-022-00795-0","url":null,"abstract":"<div><p>This paper concerns uniform continuity of real-valued functions on a (pre-)uniform frame. The aim of the paper is to characterize uniform continuity of such frame homomorphisms in terms of a farness relation between elements in the frame, and then to derive from it a separation and an extension theorem for real-valued uniform maps on uniform frames. The approach, purely order-theoretic, uses properties of the Galois maps associated with the farness relation. As a byproduct, we identify sufficient conditions under which a (continuous) scale in a frame with a preuniformity generates a real-valued uniform map.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"83 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2022-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44084751","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
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