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Algebra Universalis最新文献

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Vaughan-Lee’s nilpotent loop of size 12 is finitely based 沃恩-李的大小为12的幂零循环是基于有限的
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-11-14 DOI: 10.1007/s00012-023-00832-6
Peter Mayr
{"title":"Vaughan-Lee’s nilpotent loop of size 12 is finitely based","authors":"Peter Mayr","doi":"10.1007/s00012-023-00832-6","DOIUrl":"10.1007/s00012-023-00832-6","url":null,"abstract":"<div><p>From work of Vaughan-Lee in [12] it follows that if a finite nilpotent loop splits into a direct product of factors of prime power order, then its equational theory has a finite basis. Whether the condition on the direct decomposition is necessary has remained open since. In the same paper, Vaughan-Lee gives an explicit example of a nilpotent loop of order 12 that does not factor into loops of prime power order and asks whether it is finitely based. We give a finite basis for his example by explicitly characterizing its term functions. This also allows us to show that the subpower membership problem for this loop can be solved in polynomial time.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134878337","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
Efficient realizations of closure systems 有效实现封闭系统
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-11-11 DOI: 10.1007/s00012-023-00831-7
Robert E. Jamison
{"title":"Efficient realizations of closure systems","authors":"Robert E. Jamison","doi":"10.1007/s00012-023-00831-7","DOIUrl":"10.1007/s00012-023-00831-7","url":null,"abstract":"<div><p>As is well-known, the subalgebras of any universal algebra form an algebraic closure system. Conversely, every algebraic closure system arises as the family of subalgebras of some universal algebra, but this algebra is far from uniquely determined. This paper investigates the realization of algebraic closure systems by algebras given either by a single operation or by operations of the lowest arity. In particular, it is shown that an algebraic closure system with arity <i>n</i> in which the empty set is closed and every finitely generated closed set is countable can be realized by a single <span>((n+1))</span>-ary operation. The algebraic closure system of cosets on any group is realized by the single ternary Mal’cev term <span>(xy^{-1}z)</span>. It is shown that the closure system of cosets on an Abelian group <i>A</i> can be realized by a single binary operation if and only if <i>A</i> has at most one element of order 2. Similar results are obtained for modules over an arbitrary ring.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"85 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134796010","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
On automorphisms of categories with applications to universal algebraic geometry 范畴的自同构及其在普适代数几何中的应用
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-10-31 DOI: 10.1007/s00012-023-00829-1
Grigori Zhitomirski
{"title":"On automorphisms of categories with applications to universal algebraic geometry","authors":"Grigori Zhitomirski","doi":"10.1007/s00012-023-00829-1","DOIUrl":"10.1007/s00012-023-00829-1","url":null,"abstract":"<div><p>Let <span>({mathcal {V}})</span> be a variety of algebras of some type <span>(Omega )</span>. An interest to describing automorphisms of the category <span>(Theta ^0 ({mathcal {V}}))</span> of finitely generated free <span>({mathcal {V}})</span>-algebras was inspired by development of universal algebraic geometry founded by B. Plotkin. There are a lot of results on this subject. A common method of getting such results was suggested and applied by B. Plotkin and the author. The method is to find all terms in the language of a given variety which determine such <span>(Omega )</span>-algebras that are isomorphic to a given <span>(Theta ^0 ({mathcal {V}}))</span>-algebra and have the same underlying set with it. But this method can be applied only to automorphisms which take all objects to isomorphic ones. The aim of the present paper is to suggest another method which works in more general setting. This method is based on two main theorems. The first of them gives a general description of automorphisms of categories which are supplied with a faithful representative functor into the category of sets. The second one shows how to obtain the full description of automorphisms of the category <span>(Theta ^0 ({mathcal {V}}))</span>. This part of the paper ends with two examples. The first of them shows the preference of our method in a known situation (the variety of all semigroups) and the second one demonstrates obtaining new results (the variety of all modules over arbitrary ring with unit). The last section contains some applications to universal algebraic geometry.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134797832","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
A frame-theoretic perspective on Esakia duality Esakia对偶的一个框架理论视角
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-09-30 DOI: 10.1007/s00012-023-00827-3
G. Bezhanishvili, L. Carai, P. J. Morandi
{"title":"A frame-theoretic perspective on Esakia duality","authors":"G. Bezhanishvili,&nbsp;L. Carai,&nbsp;P. J. Morandi","doi":"10.1007/s00012-023-00827-3","DOIUrl":"10.1007/s00012-023-00827-3","url":null,"abstract":"<div><p>We introduce the category of Heyting frames, those coherent frames <i>L</i> in which the compact elements form a Heyting subalgebra of <i>L</i>, and show that it is equivalent to the category of Heyting algebras and dually equivalent to the category of Esakia spaces. This provides a frame-theoretic perspective on Esakia duality for Heyting algebras. We also generalize these results to the setting of Brouwerian algebras and Brouwerian semilattices by introducing the corresponding categories of Brouwerian frames and extending the above equivalences and dual equivalences. This provides a frame-theoretic perspective on generalized Esakia duality for Brouwerian algebras and Brouwerian semilattices.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50527785","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
On the lattice of conatural classes of linear modular lattices 关于线性模格的自然类的格
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-09-21 DOI: 10.1007/s00012-023-00828-2
Sebastián Pardo-Guerra, Hugo A. Rincón-Mejía, Manuel G. Zorrilla-Noriega, Francisco González-Bayona
{"title":"On the lattice of conatural classes of linear modular lattices","authors":"Sebastián Pardo-Guerra,&nbsp;Hugo A. Rincón-Mejía,&nbsp;Manuel G. Zorrilla-Noriega,&nbsp;Francisco González-Bayona","doi":"10.1007/s00012-023-00828-2","DOIUrl":"10.1007/s00012-023-00828-2","url":null,"abstract":"<div><p>The collection of all cohereditary classes of modules over a ring <i>R</i> is a pseudocomplemented complete big lattice. The elements of its skeleton are the conatural classes of <i>R</i>-modules. In this paper we extend some results about cohereditary classes in <i>R</i><i>-</i>Mod to the category <span>(mathcal {L_{M}})</span> of linear modular lattices, which has as objects all complete modular lattices and as morphisms all linear morphisms. We introduce the big lattice of conatural classes in <span>(mathcal {L_{M}})</span>, and we obtain some results about it, paralleling the case of <i>R</i>-Mod and arriving at its being boolean. Finally, we prove some closure properties of conatural classes in <span>(mathcal {L_{M}})</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00828-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50504465","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 variety generated by generalized subreducts of Tarski’s algebras of relations 关于Tarski关系代数的广义子导生成的多样性
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-09-01 DOI: 10.1007/s00012-023-00826-4
Dmitry A. Bredikhin
{"title":"On the variety generated by generalized subreducts of Tarski’s algebras of relations","authors":"Dmitry A. Bredikhin","doi":"10.1007/s00012-023-00826-4","DOIUrl":"10.1007/s00012-023-00826-4","url":null,"abstract":"<div><p>In the paper, a basis of identities for the variety generated by the class of groupoids that are generalized subreducts of Tarski’s algebra of relations is found. It is also proved that the corresponding class of groupoids does not form a variety.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43265355","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
Automorphisms and strongly invariant relations 自同构与强不变关系
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-08-09 DOI: 10.1007/s00012-023-00818-4
Ferdinand Börner, Martin Goldstern, Saharon Shelah
{"title":"Automorphisms and strongly invariant relations","authors":"Ferdinand Börner,&nbsp;Martin Goldstern,&nbsp;Saharon Shelah","doi":"10.1007/s00012-023-00818-4","DOIUrl":"10.1007/s00012-023-00818-4","url":null,"abstract":"<div><p>We investigate characterizations of the Galois connection <span>({{,textrm{Aut},}})</span>-<span>({{,textrm{sInv},}})</span> between sets of finitary relations on a base set <i>A</i> and their automorphisms. In particular, for <span>(A=omega _1)</span>, we construct a countable set <i>R</i> of relations that is closed under all invariant operations on relations and under arbitrary intersections, but is not closed under <span>({textrm{sInv Aut}})</span>. Our structure (<i>A</i>, <i>R</i>) has an <span>(omega )</span>-categorical first order theory. A higher order definable well-order makes it rigid, but any reduct to a finite language is homogeneous.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 4","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00818-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50464834","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
Jónsson Jónsson–Tarski algebras Jónsson-Jónsson Tarski代数
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-07-26 DOI: 10.1007/s00012-023-00824-6
Jordan DuBeau
{"title":"Jónsson Jónsson–Tarski algebras","authors":"Jordan DuBeau","doi":"10.1007/s00012-023-00824-6","DOIUrl":"10.1007/s00012-023-00824-6","url":null,"abstract":"<div><p>By studying the variety of Jónsson–Tarski algebras, we demonstrate two obstacles to the existence of large Jónsson algebras in certain varieties. First, if an algebra <i>J</i> in a language <i>L</i> has cardinality greater than <span>(|L|^+)</span> and a distributive subalgebra lattice, then it must have a proper subalgebra of size |<i>J</i>|. Second, if an algebra <i>J</i> in a language <i>L</i> satisfies <span>({{,textrm{cf},}}(|J|) &gt; 2^{|L|^+})</span> and lies in a residually small variety, then it again must have a proper subalgebra of size |<i>J</i>|. We apply the first result to show that Jónsson algebras in the variety of Jónsson–Tarski algebras cannot have cardinality greater than <span>(aleph _1)</span>. We also construct <span>(2^{aleph _1})</span> many pairwise nonisomorphic Jónsson algebras in this variety, thus proving that for some varieties the maximum possible number of Jónsson algebras can be achieved.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00824-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50515696","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 modal logic of abelian groups 阿贝尔群的模态逻辑
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-07-26 DOI: 10.1007/s00012-023-00821-9
Sören Berger, Alexander Christensen Block, Benedikt Löwe
{"title":"The modal logic of abelian groups","authors":"Sören Berger,&nbsp;Alexander Christensen Block,&nbsp;Benedikt Löwe","doi":"10.1007/s00012-023-00821-9","DOIUrl":"10.1007/s00012-023-00821-9","url":null,"abstract":"<div><p>We prove that the modal logic of abelian groups with the accessibility relation of being isomorphic to a subgroup is <span>(mathsf {S4.2})</span>.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00012-023-00821-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47568534","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
Existential relations on infinite structures 无限结构上的存在关系
IF 0.6 4区 数学
Algebra Universalis Pub Date : 2023-07-19 DOI: 10.1007/s00012-023-00819-3
Boris A. Romov
{"title":"Existential relations on infinite structures","authors":"Boris A. Romov","doi":"10.1007/s00012-023-00819-3","DOIUrl":"10.1007/s00012-023-00819-3","url":null,"abstract":"<div><p>We establish a criterion for a structure <i>M</i> on an infinite domain to have the Galois closure <span>({{,textrm{InvAut},}}(M))</span> (the set all relations on the domain of <i>M</i> that are invariant to all automorphisms of <i>M</i>) defined via infinite Boolean combinations of infinite (constructed by infinite conjunction) existential relations from <i>M</i>. Based on this approach, we present criteria for quantifier elimination in <i>M</i> via finite partial automorphisms of all existential relations from <i>M</i>, as well as criteria for (weak) homogeneity of <i>M</i>. Then we describe properties of <i>M</i> with a countable signature, for which the set of all relations, expressed by quantifier-fee formulas over <i>M</i>, is weakly inductive, that is, this set is closed under any infinitary intersection of the same arity relations. It is shown that the last condition is equivalent: for every <span>(n ge 1)</span> there are only finitely many isomorphism types for substructures of <i>M</i> generated by <i>n</i> elements. In case of algebras with a countable signature such type can be defined by the set of all solutions of a finite system of equations and inequalities produced by <i>n</i>-ary terms over those algebras. Next, we prove that for a finite <i>M</i> with a finite signature the problem of the description of any relation from <span>({{,textrm{InvAut},}}(M))</span> via the first order formula over <i>M</i>, which expresses it, is algorithmically solvable.</p></div>","PeriodicalId":50827,"journal":{"name":"Algebra Universalis","volume":"84 3","pages":""},"PeriodicalIF":0.6,"publicationDate":"2023-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46019463","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
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