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Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-31 DOI: 10.1007/s10469-025-09776-3
W. Dziobiak, M. V. Schwidefsky
{"title":"Duality for Bi-Algebraic Lattices Belonging to the Variety of (0, 1)-Lattices Generated by the Pentagon","authors":"W. Dziobiak,&nbsp;M. V. Schwidefsky","doi":"10.1007/s10469-025-09776-3","DOIUrl":"10.1007/s10469-025-09776-3","url":null,"abstract":"<p>According to G. Birkhoff, there is a categorical duality between the category of bi-algebraic distributive (0, 1)-lattices with complete (0, 1)-lattice homomorphisms as morphisms and the category of partially ordered sets with partial order-preserving maps as morphisms. We extend this classical result to the bi-algebraic lattices belonging to the variety of (0, 1)-lattices generated by the pentagon, the 5-element nonmodular lattice. Applying the extended duality, we prove that the lattice of quasivarieties contained in the variety of (0, 1)-lattices generated by the pentagon has uncountably many elements and is not distributive. This yields the following: the lattice of quasivarieties contained in a nontrivial variety of (0, 1)-lattices either is a 2-element chain or has uncountably many elements and is not distributive.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"114 - 140"},"PeriodicalIF":0.4,"publicationDate":"2025-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143109883","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
Products of Quandles
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-23 DOI: 10.1007/s10469-025-09773-6
V. G. Bardakov, D. A. Fedoseev
{"title":"Products of Quandles","authors":"V. G. Bardakov,&nbsp;D. A. Fedoseev","doi":"10.1007/s10469-025-09773-6","DOIUrl":"10.1007/s10469-025-09773-6","url":null,"abstract":"<p>We generalize the constructions of <i>Q</i>- and <i>G</i>-families of quandles introduced in the paper of A. Ishii et al. in [Ill. J. Math., <b>57</b>, No. 3, 817-838 (2013)], and establish how they are related to other constructions of quandles. A composition of structures of quandles defined on the same set is specified, and conditions are found under which this composition yields a quandle. It is proved that under such a multiplication we obtain a group that will be Abelian. Also a direct product of quandles is examined.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"75 - 97"},"PeriodicalIF":0.4,"publicationDate":"2025-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143107988","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 Simple Symplectic Groups
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-22 DOI: 10.1007/s10469-025-09774-5
Zh. Wang, W. Guo, D. V. Lytkina, V. D. Mazurov
{"title":"Periodic Groups Saturated with Finite Simple Symplectic Groups","authors":"Zh. Wang,&nbsp;W. Guo,&nbsp;D. V. Lytkina,&nbsp;V. D. Mazurov","doi":"10.1007/s10469-025-09774-5","DOIUrl":"10.1007/s10469-025-09774-5","url":null,"abstract":"<p>We study periodic groups saturated with finite simple symplectic groups.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"98 - 104"},"PeriodicalIF":0.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108905","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
Characterization of the Group A5 × A5 × A5 by the Set of Conjugacy Class Sizes
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-22 DOI: 10.1007/s10469-025-09775-4
I. B. Gorshkov, V. V. Panshin
{"title":"Characterization of the Group A5 × A5 × A5 by the Set of Conjugacy Class Sizes","authors":"I. B. Gorshkov,&nbsp;V. V. Panshin","doi":"10.1007/s10469-025-09775-4","DOIUrl":"10.1007/s10469-025-09775-4","url":null,"abstract":"<p>For a finite group<i> G</i>, we denote by <i>N</i> (<i>G</i>) the set of its conjugacy class sizes. Recently, the following question was posed: given any <i>n</i> ∈ ℕ and an arbitrary non-Abelian finite simple group <i>S</i>, is it true that <i>G</i> ≃<i> S</i><sup><i>n</i></sup> if <i>G</i> is a group with trivial center and <i>N</i> (<i>G</i>) = <i>N</i> (<i>S</i><sup><i>n</i></sup>)? The answer to this question is known for all simple groups <i>S</i> with <i>n =</i> 1, and also for <i>S</i> ∈ {<i>A</i><sub>5</sub>, <i>A</i><sub>6</sub>}, where <i>A</i><sub><i>k</i></sub> denotes the alternating group of degree <i>k</i>, with <i>n</i> = 2. It is proved that the group <i>A</i><sub>5</sub> ×<i> A</i><sub>5</sub> ×<i> A</i><sub>5</sub> is uniquely defined by the set<i> N</i> (<i>A</i><sub>5</sub> ×<i> A</i><sub>5</sub> ×<i> A</i><sub>5</sub>) in the class of finite groups with trivial center.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"105 - 113"},"PeriodicalIF":0.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108906","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
Permutation Groups and Ideals of Turing Degrees
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-22 DOI: 10.1007/s10469-025-09777-2
A. S. Morozov, V. G. Puzarenko, M. Kh. Faizrakhmanov
{"title":"Permutation Groups and Ideals of Turing Degrees","authors":"A. S. Morozov,&nbsp;V. G. Puzarenko,&nbsp;M. Kh. Faizrakhmanov","doi":"10.1007/s10469-025-09777-2","DOIUrl":"10.1007/s10469-025-09777-2","url":null,"abstract":"<p>We study degrees and degree spectra of groups <span>({mathfrak{G}}_{mathrm{I}})</span> defined on a set of permutations of the natural numbers <i>ω</i> whose degrees belong to a Turing ideal I. A necessary condition and a sufficient condition are stated which specify whether an arbitrary Turing degree belongs to the degree spectrum of a group <span>({mathfrak{G}}_{mathrm{I}})</span>. Nonprincipal ideals I for which the group <span>({mathfrak{G}}_{mathrm{I}})</span> has or does not have a degree are exemplified.</p>","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"141 - 152"},"PeriodicalIF":0.4,"publicationDate":"2025-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108999","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
Sessions of the Seminar “Algebra i Logika”
IF 0.4 3区 数学
Algebra and Logic Pub Date : 2025-01-16 DOI: 10.1007/s10469-025-09778-1
{"title":"Sessions of the Seminar “Algebra i Logika”","authors":"","doi":"10.1007/s10469-025-09778-1","DOIUrl":"10.1007/s10469-025-09778-1","url":null,"abstract":"","PeriodicalId":7422,"journal":{"name":"Algebra and Logic","volume":"63 2","pages":"153 - 154"},"PeriodicalIF":0.4,"publicationDate":"2025-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143108495","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
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
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