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Nonmatrix Varieties of Nonassociative Algebras 非结合代数的非矩阵变异
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-23 DOI: 10.1007/s10469-024-09763-0
I. P. Shestakov, V. S. Bittencourt
{"title":"Nonmatrix Varieties of Nonassociative Algebras","authors":"I. P. Shestakov,&nbsp;V. S. Bittencourt","doi":"10.1007/s10469-024-09763-0","DOIUrl":"10.1007/s10469-024-09763-0","url":null,"abstract":"<p>A variety of associative algebras is nonmatrix if it does not contain the algebra of 2 × 2 matrices over a given field. Nonmatrix varieties were introduced and studied by V. N. Latyshev in [Algebra and Logic, <b>16</b>, No. 2, 98-122 (1977); Algebra and Logic, <b>16</b>, No. 2, 122-133 (1977); Mat. Zam., <b>27</b>, No. 1, 147-156 (1980)] in connection with the Specht problem. A series of equivalent characterizations of nonmatrix varieties was obtained in [Isr. J. Math., <b>181</b>, No. 1, 337-348 (2011)]. In the present paper, the notion of nonmatrix variety is extended to nonassociative algebras, and their characterization from the last-mentioned paper is generalized to alternative, Jordan, and some other varieties of algebras.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"532 - 547"},"PeriodicalIF":0.4,"publicationDate":"2024-12-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889684","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}
引用次数: 0
Integral Classification of Endomorphisms of an Arbitrary Algebra with Finitary Operations 具有有限运算的任意代数自同态的积分分类
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-21 DOI: 10.1007/s10469-024-09769-8
A. V. Litavrin
{"title":"Integral Classification of Endomorphisms of an Arbitrary Algebra with Finitary Operations","authors":"A. V. Litavrin","doi":"10.1007/s10469-024-09769-8","DOIUrl":"10.1007/s10469-024-09769-8","url":null,"abstract":"<p>We introduce a bipolar classification with index <i>j</i> for endomorphisms of an arbitrary <i>n</i>-groupoid with <i>n</i> &gt; 1, where <i>j</i> = 1, 2, . . . , <i>n</i>. The classifications of endomorphisms constructed generalize the bipolar classification of endomorphisms of an arbitrary groupoid (i.e., a 2-groupoid) introduced previously. Using a left bipolar classification of endomorphisms of an <i>n</i>-groupoid (a particular case of the obtained classifications), we succeed in constructing an integral classification of endomorphisms of an arbitrary algebra (i.e., a structure without relations) with finitary operations.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"42 - 55"},"PeriodicalIF":0.4,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889829","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}
引用次数: 0
Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Unipotent Elements of Groups of Lie Type 关于π-根的一个Sharp Baer-Suzuki定理：Lie型群的单幂元
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-21 DOI: 10.1007/s10469-024-09760-3
A-M. Liu, Zh. Wang, D. O. Revin
{"title":"Toward a Sharp Baer–Suzuki Theorem for the π-Radical: Unipotent Elements of Groups of Lie Type","authors":"A-M. Liu,&nbsp;Zh. Wang,&nbsp;D. O. Revin","doi":"10.1007/s10469-024-09760-3","DOIUrl":"10.1007/s10469-024-09760-3","url":null,"abstract":"<p>We will look into the following conjecture, which, if valid, would allow us to formulate an unimprovable analog of the Baer–Suzuki theorem for the <i>π</i>-radical of a finite group (here <i>π</i> is an arbitrary set of primes). For an odd prime number <i>r</i>, put <i>m = r</i>, if <i>r =</i> 3, and <i>m = r</i> - 1 if <i>r</i> ≥ 5. Let L be a simple non-Abelian group whose order has a prime divisor <i>s</i> such that <i>s</i> = <i>r</i> if <i>r</i> divides |<i>L</i>| and <i>s</i> &gt; <i>r</i> otherwise. Suppose also that <i>x</i> is an automorphism of prime order of <i>L</i>. Then some m conjugates of <i>x</i> in the group <span>(langle L,xrangle )</span> generate a subgroup of order divisible by <i>s</i>. The conjecture is confirmed for the case where <i>L</i> is a group of Lie type and <i>x</i> is an automorphism induced by a unipotent element.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"476 - 500"},"PeriodicalIF":0.4,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889826","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}
引用次数: 0
Periodic Groups Saturated with Finite Frobenius Groups with Complements of Orders Divisible by a Prime Number 具有可被一个素数整除的补阶的有限Frobenius群饱和的周期群
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-21 DOI: 10.1007/s10469-024-09759-w
B. E. Durakov
{"title":"Periodic Groups Saturated with Finite Frobenius Groups with Complements of Orders Divisible by a Prime Number","authors":"B. E. Durakov","doi":"10.1007/s10469-024-09759-w","DOIUrl":"10.1007/s10469-024-09759-w","url":null,"abstract":"<p>A finite Frobenius group in which the order of complements is divisible by a prime number <i>p</i> is called a Φ<sub><i>p</i></sub>-group. We prove the theorem stating the following. Let <i>G</i> be a periodic group with a finite element a of prime order <i>p &gt;</i> 2 saturated with Φ<sub><i>p</i></sub>-groups. Then <i>G</i> = <i>F λ H</i> is a Frobenius group with kernel <i>F</i> and complement <i>H</i>. If <i>G</i> contains an involution <i>i</i> commuting with the element a, then <i>H = C</i><sub><i>G</i></sub>(<i>i</i>) and <i>F</i> is Abelian, and <i>H = N</i><sub><i>G</i></sub>(<span>(langle arangle )</span>) otherwise.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"62 6","pages":"471 - 475"},"PeriodicalIF":0.4,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889827","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}
引用次数: 0
Box-Quasimetrics and Horizontal Joinability on Cartan Groups Cartan群上的盒拟测度和水平接合性
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-21 DOI: 10.1007/s10469-024-09767-w
A. V. Greshnov, V. S. Kostyrkin
{"title":"Box-Quasimetrics and Horizontal Joinability on Cartan Groups","authors":"A. V. Greshnov,&nbsp;V. S. Kostyrkin","doi":"10.1007/s10469-024-09767-w","DOIUrl":"10.1007/s10469-024-09767-w","url":null,"abstract":"<p>On a Cartan group <span>({mathbb{K}})</span> equipped with a Carnot–Carathéodory metric <i>d</i><sub><i>cc</i></sub>, we find the exact value of a constant in the (1, <i>q</i><sub>2</sub>)-generalized triangle inequality for its Box-quasimetric. It is proved that any two points <i>x</i>, <i>y</i> ∈ <span>({mathbb{K}})</span> can be joined by a horizontal <i>k</i>-broken line <span>({L}_{x,y}^{k})</span>, <i>k</i> ≤ 6; moreover, the length of such a broken line <span>({L}_{x,y}^{k})</span> does not exceed the quantity <i>Cd</i><sub><i>cc</i></sub>(<i>x</i>, <i>y</i>) for some constant <i>C</i> not depending on the choice of <i>x</i>, <i>y</i> ∈ <span>({mathbb{K}})</span>. The value 6 here is nearly optimal.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"10 - 20"},"PeriodicalIF":0.4,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889825","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}
引用次数: 0
Residual π-Finiteness of Tubular Groups 管群的残差π有限性
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-21 DOI: 10.1007/s10469-024-09768-9
F. A. Dudkin, A. V. Usikov
{"title":"Residual π-Finiteness of Tubular Groups","authors":"F. A. Dudkin,&nbsp;A. V. Usikov","doi":"10.1007/s10469-024-09768-9","DOIUrl":"10.1007/s10469-024-09768-9","url":null,"abstract":"<p>A finitely generated group<i> G</i>, which acts on a tree so that all edge stabilizers are infinite cyclic groups and all vertex stabilizers are free rank 2 Abelian groups, is called a tubular group. Every tubular group is isomorphic to the fundamental group <i>π</i><sub>1</sub>(𝒢) of a suitable finite graph 𝒢 of groups. We prove a criterion for residual <i>π</i>-finiteness of tubular groups presented by trees of groups. Also we state a criterion for residual p-finiteness of tubular groups whose corresponding graph contains one edge outside a maximal subtree.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 1","pages":"28 - 41"},"PeriodicalIF":0.4,"publicationDate":"2024-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142889828","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}
引用次数: 0
Levi Classes of Quasivarieties of Nilpotent Groups of Class at Most Two 最多两个类的幂零群的拟变异的李维类
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-19 DOI: 10.1007/s10469-024-09761-2
S. A. Shakhova
{"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}
引用次数: 0
The Length of an Unsatisfiable Subformula 不可满足子公式的长度
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-19 DOI: 10.1007/s10469-024-09771-0
A. V. Seliverstov
{"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}
引用次数: 0
Some Isomorphisms between Incidence Algebras and Group Algebras 关联代数与群代数之间的若干同构
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-17 DOI: 10.1007/s10469-024-09766-x
I. V. Dudin, P. A. Krylov
{"title":"Some Isomorphisms between Incidence Algebras and Group Algebras","authors":"I. V. Dudin,&nbsp;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}
引用次数: 0
Existence of Independent Quasi-Equational Bases. II 独立拟方程基的存在性。2
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2024-12-17 DOI: 10.1007/s10469-024-09762-1
M. V. Schwidefsky
{"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}
引用次数: 0
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