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Decomposition of (hbox {d}_{text {0}})-algebras (hbox{d}_{text{0}})-代数的分解
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-09-24 DOI: 10.1007/s00012-024-00871-7
Anna Avallone, Paolo Vitolo
{"title":"Decomposition of (hbox {d}_{text {0}})-algebras","authors":"Anna Avallone,&nbsp;Paolo Vitolo","doi":"10.1007/s00012-024-00871-7","DOIUrl":"10.1007/s00012-024-00871-7","url":null,"abstract":"<div><p>We generalize to <span>(hbox {d}_{text {0}})</span>-algebras a result of Riečanová about the decomposition of a D-lattice by means of a family of central elements.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413607","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
Nilpotent groups in lattice framework 网格框架中的无势群
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-09-23 DOI: 10.1007/s00012-024-00873-5
Jelena Jovanović, Branimir Šešelja, Andreja Tepavčević
{"title":"Nilpotent groups in lattice framework","authors":"Jelena Jovanović,&nbsp;Branimir Šešelja,&nbsp;Andreja Tepavčević","doi":"10.1007/s00012-024-00873-5","DOIUrl":"10.1007/s00012-024-00873-5","url":null,"abstract":"<div><p>In the framework of weak congruence lattices, many classes of groups have been characterized up to now, in completely lattice-theoretic terms. In this note, the center of the group is captured lattice-theoretically and nilpotent groups are characterized by lattice properties.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142413279","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
Odd and even Fibonacci lattices arising from a Garside monoid 由加塞德单体产生的奇数和偶数斐波那契网格
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-09-17 DOI: 10.1007/s00012-024-00867-3
Thomas Gobet, Baptiste Rognerud
{"title":"Odd and even Fibonacci lattices arising from a Garside monoid","authors":"Thomas Gobet,&nbsp;Baptiste Rognerud","doi":"10.1007/s00012-024-00867-3","DOIUrl":"10.1007/s00012-024-00867-3","url":null,"abstract":"<div><p>We study two families of lattices whose number of elements are given by the numbers in even (respectively odd) positions in the Fibonacci sequence. The even Fibonacci lattice arises as the lattice of simple elements of a Garside monoid partially ordered by left-divisibility, and the odd Fibonacci lattice is an order ideal in the even one. We give a combinatorial proof of the lattice property, relying on a description of words for the Garside element in terms of Schröder trees, and on a recursive description of the even Fibonacci lattice. This yields an explicit formula to calculate meets and joins in the lattice. As a byproduct we also obtain that the number of words for the Garside element is given by a little Schröder number.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142250339","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
Cartesian closed varieties I: the classification theorem 笛卡儿封闭变体 I：分类定理
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-09-13 DOI: 10.1007/s00012-024-00869-1
Richard Garner
{"title":"Cartesian closed varieties I: the classification theorem","authors":"Richard Garner","doi":"10.1007/s00012-024-00869-1","DOIUrl":"10.1007/s00012-024-00869-1","url":null,"abstract":"<div><p>In 1990, Johnstone gave a syntactic characterisation of the equational theories whose associated varieties are cartesian closed. Among such theories are all <i>unary</i> theories—whose models are sets equipped with an action by a monoid <i>M</i>—and all <i>hyperaffine</i> theories—whose models are sets with an action by a Boolean algebra <i>B</i>. We improve on Johnstone’s result by showing that an equational theory is cartesian closed just when its operations have a unique hyperaffine–unary decomposition. It follows that any non-degenerate cartesian closed variety is a variety of sets equipped with compatible actions by a monoid <i>M</i> and a Boolean algebra <i>B</i>; this is the classification theorem of the title.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00869-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202457","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
Natural dualities for varieties generated by finite positive MV-chains 有限正 MV 链生成的品种的自然对偶性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-09-06 DOI: 10.1007/s00012-024-00868-2
Wolfgang Poiger
{"title":"Natural dualities for varieties generated by finite positive MV-chains","authors":"Wolfgang Poiger","doi":"10.1007/s00012-024-00868-2","DOIUrl":"10.1007/s00012-024-00868-2","url":null,"abstract":"<div><p>We provide a simple natural duality for the varieties generated by the negation- and implication-free reduct of a finite MV-chain. We study these varieties through the dual equivalences thus obtained. For example, we fully characterize their algebraically closed, existentially closed and injective members. We also explore the relationship between this natural duality and Priestley duality in terms of distributive skeletons and Priestley powers.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202458","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
Quasivarieties of algebras whose compact relative congruences are principal 其紧凑相对全等为主项的代数准变量
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-08-27 DOI: 10.1007/s00012-024-00866-4
Anvar M. Nurakunov
{"title":"Quasivarieties of algebras whose compact relative congruences are principal","authors":"Anvar M. Nurakunov","doi":"10.1007/s00012-024-00866-4","DOIUrl":"10.1007/s00012-024-00866-4","url":null,"abstract":"<div><p>A quasivariety <span>(mathfrak N)</span> is called <i>relative congruence principal</i> if, for every algebra <span>(Ain mathfrak N)</span>, every compact <span>(mathfrak N)</span>-congruence on <i>A</i> is a principal <span>(mathfrak N)</span>-congruence. We characterize relative congruence principal quasivarieties in terms of one identity and two quasi-identities. We will use the characterization to show that there exists a continuum of relative congruence principal quasivarieties of algebras of a signature <span>(sigma )</span>, provided <span>(sigma )</span> contains at least one operation of arity greater than 1. Several examples are provided.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202460","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
Override and restricted union for partial functions 部分函数的覆盖和受限联合
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-08-12 DOI: 10.1007/s00012-024-00864-6
Tim Stokes
{"title":"Override and restricted union for partial functions","authors":"Tim Stokes","doi":"10.1007/s00012-024-00864-6","DOIUrl":"10.1007/s00012-024-00864-6","url":null,"abstract":"<div><p>The <i>override</i> operation <span>(sqcup )</span> is a natural one in computer science, and has connections with other areas of mathematics such as hyperplane arrangements. For arbitrary functions <i>f</i> and <i>g</i>, <span>(fsqcup g)</span> is the function with domain <span>({{,textrm{dom},}}(f)cup {{,textrm{dom},}}(g))</span> that agrees with <i>f</i> on <span>({{,textrm{dom},}}(f))</span> and with <i>g</i> on <span>({{,textrm{dom},}}(g) backslash {{,textrm{dom},}}(f))</span>. Jackson and the author have shown that there is no finite axiomatisation of algebras of functions of signature <span>((sqcup ))</span>. But adding operations (such as <i>update</i>) to this minimal signature can lead to finite axiomatisations. For the functional signature <span>((sqcup ,backslash ))</span> where <span>(backslash )</span> is set-theoretic difference, Cirulis has given a finite equational axiomatisation as subtraction o-semilattices. Define <span>(fcurlyvee g=(fsqcup g)cap (gsqcup f))</span> for all functions <i>f</i> and <i>g</i>; this is the largest domain restriction of the binary relation <span>(fcup g)</span> that gives a partial function. Now <span>(fcap g=fbackslash (fbackslash g))</span> and <span>(fsqcup g=fcurlyvee (fcurlyvee g))</span> for all functions <i>f</i>, <i>g</i>, so the signatures <span>((curlyvee ))</span> and <span>((sqcup ,cap ))</span> are both intermediate between <span>((sqcup ))</span> and <span>((sqcup ,backslash ))</span> in expressive power. We show that each is finitely axiomatised, with the former giving a proper quasivariety and the latter the variety of associative distributive o-semilattices in the sense of Cirulis.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00864-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202461","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
(varvec{S})-preclones and the Galois connection (varvec{{}^{S}{}textrm{Pol}})–(varvec{{}^{S}{}textrm{Inv}}), Part I $$varvec{S}$ -preclones and Galois connection $$varvec{{}^{S}{}textrm{Pol}}$ - $$varvec{{}^{S}{}textrm{Inv}}$, Part I
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-07-09 DOI: 10.1007/s00012-024-00863-7
Peter Jipsen, Erkko Lehtonen, Reinhard Pöschel
{"title":"(varvec{S})-preclones and the Galois connection (varvec{{}^{S}{}textrm{Pol}})–(varvec{{}^{S}{}textrm{Inv}}), Part I","authors":"Peter Jipsen,&nbsp;Erkko Lehtonen,&nbsp;Reinhard Pöschel","doi":"10.1007/s00012-024-00863-7","DOIUrl":"10.1007/s00012-024-00863-7","url":null,"abstract":"<div><p>We consider <i>S</i>-<i>operations</i> <span>(f :A^{n} rightarrow A)</span> in which each argument is assigned a <i>signum</i> <span>(s in S)</span> representing a “property” such as being order-preserving or order-reversing with respect to a fixed partial order on <i>A</i>. The set <i>S</i> of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of <i>S</i>-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all <i>S</i>-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of <i>S</i>-<i>preclone</i>. We introduce <i>S</i>-<i>relations</i> <span>(varrho = (varrho _{s})_{s in S})</span>, <i>S</i>-<i>relational clones</i>, and a preservation property (<img>), and we consider the induced Galois connection <span>({}^{S}{}textrm{Pol})</span>–<span>({}^{S}{}textrm{Inv})</span>. The <i>S</i>-preclones and <i>S</i>-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all <i>S</i>-preclones on <i>A</i>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00863-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568120","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
On the networks of large embeddings 关于大型嵌入网络
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-07-09 DOI: 10.1007/s00012-024-00862-8
Tuğba Aslan, Mohamed Khaled, Gergely Székely
{"title":"On the networks of large embeddings","authors":"Tuğba Aslan,&nbsp;Mohamed Khaled,&nbsp;Gergely Székely","doi":"10.1007/s00012-024-00862-8","DOIUrl":"10.1007/s00012-024-00862-8","url":null,"abstract":"<div><p>We define a special network that exhibits the large embeddings in any class of similar algebras. With the aid of this network, we introduce a notion of distance that conceivably counts the minimum number of dissimilarities, in a sense, between two given algebras in the class in hand; with the possibility that this distance may take the value <span>(infty )</span>. We display a number of inspirational examples from different areas of algebra, e.g., group theory and monounary algebras, to show that this research direction can be quite remarkable.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00862-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141568116","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 Freudenthal and other compactifications of continuous frames 弗赖登塔尔法和其他连续框架压缩法
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2024-07-04 DOI: 10.1007/s00012-024-00857-5
Simo Mthethwa, Gugulethu Nogwebela
{"title":"The Freudenthal and other compactifications of continuous frames","authors":"Simo Mthethwa,&nbsp;Gugulethu Nogwebela","doi":"10.1007/s00012-024-00857-5","DOIUrl":"10.1007/s00012-024-00857-5","url":null,"abstract":"<div><p>The <i>N</i>-star compactifications of frames are the frame-theoretic counterpart of the <i>N</i>-point compactifications of locally compact Hausdorff spaces. A <span>(pi )</span>-compactification of a frame <i>L</i> is a compactification constructed using a special type of a basis called a <span>(pi )</span>-compact basis; the Freudenthal compactification is the largest <span>(pi )</span>-compactification of a rim-compact frame. As one of the main results, we show that the Freudenthal compactification of a regular continuous frame is the least upper bound for the set of all <i>N</i>-star compactifications. A compactification whose right adjoint preserves disjoint binary joins is called perfect. We establish a class of frames for which <i>N</i>-star compactifications are always perfect. For the class of zero-dimensional frames, we construct a compactification which is isomorphic to the Banaschewski compactification and the Freudenthal compactification; in some special case, this compactification is isomorphic to the Stone–Čech compactification.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-024-00857-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141552278","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
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