Pub Date : 2024-12-19DOI: 10.1007/s10469-024-09761-2
S. A. Shakhova
A Levi class (Lleft(mathcal{M}right)) generated by a class (left(mathcal{M}right)) of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to (left(mathcal{M}right)). Let p be a prime and p ≠ 2, let Hp be a free group of rank 2 in the variety of nilpotent groups of class at most 2 with commutator subgroup of exponent p, and let qHp be the quasivariety generated by the group Hp. It is shown that there exists a set of quasivarieties (mathcal{M}) of cardinality continuum such that (Lleft(mathcal{M}right)) = L(qHp). Let s be a natural number, s ≥ 2. We specify a system of quasi-identities defining L(q(Hp, ({Z}_{{p}^{s}}))), and prove that there exists a set of quasivarieties (mathcal{M}) of cardinality continuum such that (Lleft(mathcal{M}right)) = L(q(Hp, ({Z}_{{p}^{s}}))), where ({Z}_{{p}^{s}}) is a cyclic group of order ps; q(Hp, ({Z}_{{p}^{s}})) is the quasivariety generated by the groups Hp and ({Z}_{{p}^{s}}.)
{"title":"Levi Classes of Quasivarieties of Nilpotent Groups of Class at Most Two","authors":"S. A. Shakhova","doi":"10.1007/s10469-024-09761-2","DOIUrl":"10.1007/s10469-024-09761-2","url":null,"abstract":"<p>A Levi class <span>(Lleft(mathcal{M}right))</span> generated by a class <span>(left(mathcal{M}right))</span> of groups is the class of all groups in which the normal closure of every cyclic subgroup belongs to <span>(left(mathcal{M}right))</span>. Let p be a prime and <i>p</i> ≠ 2, let <i>H</i><sub><i>p</i></sub> be a free group of rank 2 in the variety of nilpotent groups of class at most 2 with commutator subgroup of exponent <i>p</i>, and let <i>qH</i><sub><i>p</i></sub> be the quasivariety generated by the group <i>H</i><sub><i>p</i></sub>. It is shown that there exists a set of quasivarieties <span>(mathcal{M})</span> of cardinality continuum such that <span>(Lleft(mathcal{M}right))</span> = <i>L</i>(<i>qH</i><sub><i>p</i></sub>). Let <i>s</i> be a natural number, <i>s</i> ≥ 2. We specify a system of quasi-identities defining <i>L</i>(<i>q</i>(<i>H</i><sub><i>p</i></sub>, <span>({Z}_{{p}^{s}})</span>)), and prove that there exists a set of quasivarieties <span>(mathcal{M})</span> of cardinality continuum such that <span>(Lleft(mathcal{M}right))</span> = <i>L</i>(<i>q</i>(<i>H</i><sub><i>p</i></sub>, <span>({Z}_{{p}^{s}})</span>)), where <span>({Z}_{{p}^{s}})</span> is a cyclic group of order <i>p</i><sup><i>s</i></sup>; <i>q</i>(<i>H</i><sub><i>p</i></sub>, <span>({Z}_{{p}^{s}})</span>) is the quasivariety generated by the groups <i>H</i><sub><i>p</i></sub> and <span>({Z}_{{p}^{s}}.)</span></p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"501 - 515"},"PeriodicalIF":0.4,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-19DOI: 10.1007/s10469-024-09771-0
A. V. Seliverstov
We find a bound for the length of a conjunction of some propositional formulas, for which every unsatisfiable formula contains an unsatisfiable subformula. In particular, this technique applies to formulas in conjunctive normal form with restrictions on the number of true literals within every elementary disjunction, as well as for 2-CNFs, for symmetric 3-CNFs, and for conjunctions of voting functions in three literals. A lower bound on the rank of some matrices is used in proofs.
{"title":"The Length of an Unsatisfiable Subformula","authors":"A. V. Seliverstov","doi":"10.1007/s10469-024-09771-0","DOIUrl":"10.1007/s10469-024-09771-0","url":null,"abstract":"<p>We find a bound for the length of a conjunction of some propositional formulas, for which every unsatisfiable formula contains an unsatisfiable subformula. In particular, this technique applies to formulas in conjunctive normal form with restrictions on the number of true literals within every elementary disjunction, as well as for 2-CNFs, for symmetric 3-CNFs, and for conjunctions of voting functions in three literals. A lower bound on the rank of some matrices is used in proofs.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"65 - 72"},"PeriodicalIF":0.4,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1007/s10469-024-09766-x
I. V. Dudin, P. A. Krylov
Relations between some constructions based on incidence rings and group rings are considered.
考虑了一些基于关联环和群环的结构之间的关系。
{"title":"Some Isomorphisms between Incidence Algebras and Group Algebras","authors":"I. V. Dudin, P. A. Krylov","doi":"10.1007/s10469-024-09766-x","DOIUrl":"10.1007/s10469-024-09766-x","url":null,"abstract":"<p>Relations between some constructions based on incidence rings and group rings are considered.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"21 - 27"},"PeriodicalIF":0.4,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1007/s10469-024-09762-1
M. V. Schwidefsky
If a certain condition holds for a quasivariety K then K contains continuum many subquasivarieties having a finitely partitionable ω-independent quasi-equational basis relative to K. This is true, in particular, for each almost ff-universal quasivariety K.
{"title":"Existence of Independent Quasi-Equational Bases. II","authors":"M. V. Schwidefsky","doi":"10.1007/s10469-024-09762-1","DOIUrl":"10.1007/s10469-024-09762-1","url":null,"abstract":"<p>If a certain condition holds for a quasivariety <b>K</b> then <b>K</b> contains continuum many subquasivarieties having a finitely partitionable ω-independent quasi-equational basis relative to <b>K</b>. This is true, in particular, for each almost ff-universal quasivariety <b>K</b>.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"516 - 531"},"PeriodicalIF":0.4,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-17DOI: 10.1007/s10469-024-09770-1
S. V. Pchelintsev, I. P. Shestakov
It is proved that, for any natural n, the subalgebra generated by words of length divisible by n on generators (the Veronese n-subalgebra) in a free finitely generated alternative algebra is finitely generated.
{"title":"Finite Generatedness of Veronese Subalgebras of a Free Alternative Algebra of Finite Rank","authors":"S. V. Pchelintsev, I. P. Shestakov","doi":"10.1007/s10469-024-09770-1","DOIUrl":"10.1007/s10469-024-09770-1","url":null,"abstract":"<p>It is proved that, for any natural <i>n</i>, the subalgebra generated by words of length divisible by <i>n</i> on generators (the Veronese <i>n</i>-subalgebra) in a free finitely generated alternative algebra is finitely generated.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"56 - 64"},"PeriodicalIF":0.4,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890526","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-16DOI: 10.1007/s10469-024-09765-y
A. I. Budkin
It is proved that if G is a group without elements of order 2, and the normal closure of every 2-generated subgroup of G is a nilpotent group of class at most 3, then G will be a nilpotent group of class at most 4. It is also shown that the restriction on second-order elements cannot be lifted.
{"title":"Groups with Restrictions on Normal Subgroups","authors":"A. I. Budkin","doi":"10.1007/s10469-024-09765-y","DOIUrl":"10.1007/s10469-024-09765-y","url":null,"abstract":"<p>It is proved that if <i>G</i> is a group without elements of order 2, and the normal closure of every 2-generated subgroup of <i>G</i> is a nilpotent group of class at most 3, then <i>G</i> will be a nilpotent group of class at most 4. It is also shown that the restriction on second-order elements cannot be lifted.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"1 - 9"},"PeriodicalIF":0.4,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142890519","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-16DOI: 10.1007/s10469-024-09764-z
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-024-09764-z","DOIUrl":"10.1007/s10469-024-09764-z","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"548 - 551"},"PeriodicalIF":0.4,"publicationDate":"2024-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-12-14DOI: 10.1007/s10469-024-09772-z
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-024-09772-z","DOIUrl":"10.1007/s10469-024-09772-z","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"73 - 74"},"PeriodicalIF":0.4,"publicationDate":"2024-12-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-04DOI: 10.1007/s10469-024-09757-y
A. N. Shevlyakov
We prove that the wreath product C = A ≀ B of a semigroup A with zero and an infinite cyclic semigroup B is qω-compact (logically Noetherian). Our result partially solves I. Plotkin‘s problem for wreath products.
我们证明，有零的半群 A 和无限循环半群 B 的花环积 C = A ≀ B 是 qω-compact 的（逻辑上是诺特的）。我们的结果部分解决了 I. Plotkin 的花环积问题。
{"title":"Wreath Products of Semigroups and Plotkin’s Problem","authors":"A. N. Shevlyakov","doi":"10.1007/s10469-024-09757-y","DOIUrl":"10.1007/s10469-024-09757-y","url":null,"abstract":"<p>We prove that the wreath product <i>C</i> = <i>A</i> ≀ <i>B</i> of a semigroup A with zero and an infinite cyclic semigroup B is <b>q</b><sub><i>ω</i></sub>-compact (logically Noetherian). Our result partially solves I. Plotkin‘s problem for wreath products.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 5","pages":"448 - 467"},"PeriodicalIF":0.4,"publicationDate":"2024-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409889","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-02DOI: 10.1007/s10469-024-09753-2
M. A. Vsemirnov, Ya. N. Nuzhin
It is proved that among finite simple non-Abelian groups only the groups U3(3) and A8 are not generated by three conjugate involutions. This result is obtained modulo a known conjecture on the description of finite simple groups generated by two elements of orders 2 and 3.
{"title":"Generating Triples of Conjugate Involutions for Finite Simple Groups","authors":"M. A. Vsemirnov, Ya. N. Nuzhin","doi":"10.1007/s10469-024-09753-2","DOIUrl":"10.1007/s10469-024-09753-2","url":null,"abstract":"<p>It is proved that among finite simple non-Abelian groups only the groups <i>U</i><sub>3</sub>(3) and <i>A</i><sub>8</sub> are not generated by three conjugate involutions. This result is obtained modulo a known conjecture on the description of finite simple groups generated by two elements of orders 2 and 3.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 5","pages":"379 - 397"},"PeriodicalIF":0.4,"publicationDate":"2024-10-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142409491","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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