Pub Date : 2025-11-01Epub Date: 2025-09-05DOI: 10.1016/j.jpaa.2025.108083
Kürşat Sözer , Alexis Virelizier
Given a crossed module χ, we introduce Hopf χ-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and χ-graded (meaning that both objects and morphisms have degrees which are related via χ).
{"title":"Hopf crossed module (co)algebras","authors":"Kürşat Sözer , Alexis Virelizier","doi":"10.1016/j.jpaa.2025.108083","DOIUrl":"10.1016/j.jpaa.2025.108083","url":null,"abstract":"<div><div>Given a crossed module <em>χ</em>, we introduce Hopf <em>χ</em>-(co)algebras which generalize Hopf algebras and Hopf group-(co)algebras. We interpret them as Hopf algebras in some symmetric monoidal category. We prove that their categories of representations are monoidal and <em>χ</em>-graded (meaning that both objects and morphisms have degrees which are related via <em>χ</em>).</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108083"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145027711","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-10-01DOI: 10.1016/j.jpaa.2025.108105
Sophie Raynor
Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.
{"title":"Functorial, operadic and modular operadic combinatorics of circuit algebras","authors":"Sophie Raynor","doi":"10.1016/j.jpaa.2025.108105","DOIUrl":"10.1016/j.jpaa.2025.108105","url":null,"abstract":"<div><div>Circuit algebras are a symmetric analogue of Jones's planar algebras introduced to study finite-type invariants of virtual knotted objects. Circuit algebra structures appear, in different forms, across mathematics. This paper provides a dictionary for translating between their diverse incarnations and describing their wider context. A formal definition of a broad class of circuit algebras is established and three equivalent descriptions of circuit algebras are provided: in terms of operads of wiring diagrams, modular operads and categories of Brauer diagrams. As an application, circuit algebra characterisations of algebras over the orthogonal and symplectic groups are given.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108105"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145268346","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-10-21DOI: 10.1016/j.jpaa.2025.108110
Giacomo Tendas
Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally Φ-presentable -categories for a sound class Φ, (2) the problem of whether every Φ-accessible -category is Ψ-accessible, for given sound classes , and (3) a notion of Φ-ary equational theory whose -categories of models characterize algebras for Φ-ary monads on .
{"title":"More on soundness in the enriched context","authors":"Giacomo Tendas","doi":"10.1016/j.jpaa.2025.108110","DOIUrl":"10.1016/j.jpaa.2025.108110","url":null,"abstract":"<div><div>Working within enriched category theory, we further develop the use of soundness, introduced by Adámek, Borceux, Lack, and Rosický for ordinary categories. In particular we investigate: (1) the theory of locally Φ-presentable <span><math><mi>V</mi></math></span>-categories for a sound class Φ, (2) the problem of whether every Φ-accessible <span><math><mi>V</mi></math></span>-category is Ψ-accessible, for given sound classes <span><math><mi>Φ</mi><mo>⊆</mo><mi>Ψ</mi></math></span>, and (3) a notion of Φ-ary equational theory whose <span><math><mi>V</mi></math></span>-categories of models characterize algebras for Φ-ary monads on <span><math><mi>V</mi></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108110"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145363618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-10-01DOI: 10.1016/j.jpaa.2025.108106
Lee Tae Young
We determine precisely which irreducible hypergeometric sheaves have an extraspecial normalizer in characteristic 2 as their geometric monodromy groups. This resolves the last open case of the classification of local monodromy at 0 of irreducible hypergeometric sheaves with finite geometric monodromy group.
{"title":"Hypergeometric sheaves and extraspecial groups in even characteristic","authors":"Lee Tae Young","doi":"10.1016/j.jpaa.2025.108106","DOIUrl":"10.1016/j.jpaa.2025.108106","url":null,"abstract":"<div><div>We determine precisely which irreducible hypergeometric sheaves have an extraspecial normalizer in characteristic 2 as their geometric monodromy groups. This resolves the last open case of the classification of local monodromy at 0 of irreducible hypergeometric sheaves with finite geometric monodromy group.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108106"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-09-25DOI: 10.1016/j.jpaa.2025.108099
Henry Bradford
In [4] Bou-Rabee and Seward constructed examples of finitely generated residually finite groups G whose residual finiteness growth function can be at least as fast as any prescribed function. In this note we describe a modified version of their construction, which allows us to give a complementary upper bound on . As such, every nondecreasing function at least is close to the residual finiteness growth function of some finitely generated group. We also have similar result for the full residual finiteness growth function and for the divisibility function.
{"title":"On the spectrum of residual finiteness growth functions","authors":"Henry Bradford","doi":"10.1016/j.jpaa.2025.108099","DOIUrl":"10.1016/j.jpaa.2025.108099","url":null,"abstract":"<div><div>In <span><span>[4]</span></span> Bou-Rabee and Seward constructed examples of finitely generated residually finite groups <em>G</em> whose residual finiteness growth function <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span> can be at least as fast as any prescribed function. In this note we describe a modified version of their construction, which allows us to give a complementary upper bound on <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>. As such, every nondecreasing function at least <span><math><mi>exp</mi><mo></mo><mo>(</mo><mi>n</mi><mi>log</mi><mo></mo><msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mi>log</mi><mo></mo><mi>log</mi><mo></mo><msup><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></span> is close to the residual finiteness growth function of some finitely generated group. We also have similar result for the <em>full</em> residual finiteness growth function and for the divisibility function.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108099"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145222118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-09-09DOI: 10.1016/j.jpaa.2025.108085
Marino Gran , Thomas Letourmy , Leandro Vendramin
The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings–Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.
{"title":"Hopf formulae for homology of skew braces","authors":"Marino Gran , Thomas Letourmy , Leandro Vendramin","doi":"10.1016/j.jpaa.2025.108085","DOIUrl":"10.1016/j.jpaa.2025.108085","url":null,"abstract":"<div><div>The variety of skew braces contains several interesting subcategories as subvarieties, as for instance the varieties of radical rings, of groups and of abelian groups. In this article the methods of non-abelian homological algebra are applied to establish some new Hopf formulae for homology of skew braces, where the coefficient functors are the reflectors from the variety of skew braces to each of the three above-mentioned subvarieties. The corresponding central extensions of skew braces are characterized in purely algebraic terms, leading to some new results, such as an explicit Stallings–Stammbach exact sequence associated with any exact sequence of skew braces, and a new result concerning central series.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108085"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145050626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-10-10DOI: 10.1016/j.jpaa.2025.108108
Christopher A. Schroeder, Hung P. Tong-Viet
A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As an application, we obtain some new 2-nilpotency criteria.
{"title":"Finite groups all of whose maximal subgroups have almost odd index","authors":"Christopher A. Schroeder, Hung P. Tong-Viet","doi":"10.1016/j.jpaa.2025.108108","DOIUrl":"10.1016/j.jpaa.2025.108108","url":null,"abstract":"<div><div>A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As an application, we obtain some new 2-nilpotency criteria.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108108"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145333348","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-09-10DOI: 10.1016/j.jpaa.2025.108090
Jun Hu , Huansheng Li , Shuo Li
In this paper, we use the cyclotomic Mackey decomposition and branching rules of the seminormal bases of the semisimple cyclotomic Hecke algebras of type to give a new approach to computing the Schur element of for each ℓ-partition . Our formulae give a simple recursive relation between the Schur element and the Schur element of , where . We give our main results for both the non-degenerate and the degenerate cyclotomic Hecke algebras of type . The formulae of the Schur element that we derived are different superficially from all the known formulae in the literature.
{"title":"New formulae for the Schur elements of the cyclotomic Hecke algebra of type G(ℓ,1,n)","authors":"Jun Hu , Huansheng Li , Shuo Li","doi":"10.1016/j.jpaa.2025.108090","DOIUrl":"10.1016/j.jpaa.2025.108090","url":null,"abstract":"<div><div>In this paper, we use the cyclotomic Mackey decomposition and branching rules of the seminormal bases of the semisimple cyclotomic Hecke algebras of type <span><math><mi>G</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span> to give a new approach to computing the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> for each <em>ℓ</em>-partition <span><math><mi>λ</mi><mo>∈</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. Our formulae give a simple recursive relation between the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>λ</mi></mrow></msub></math></span> and the Schur element <span><math><msub><mrow><mi>s</mi></mrow><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub></mrow></msub></math></span> of <span><math><msub><mrow><mi>H</mi></mrow><mrow><mi>ℓ</mi><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span>, where <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>:</mo><mo>=</mo><mi>Shape</mi><mo>(</mo><msub><mrow><mo>(</mo><msup><mrow><mi>t</mi></mrow><mrow><mi>λ</mi></mrow></msup><mo>)</mo></mrow><mrow><mo>↓</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow></msub><mo>)</mo></math></span>. We give our main results for both the non-degenerate and the degenerate cyclotomic Hecke algebras of type <span><math><mi>G</mi><mo>(</mo><mi>ℓ</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo>)</mo></math></span>. The formulae of the Schur element that we derived are different superficially from all the known formulae in the literature.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108090"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145061192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-09-15DOI: 10.1016/j.jpaa.2025.108094
Shripad M. Garge , Deep H. Makadiya
Let R be a commutative ring with unity. Consider the twisted Chevalley group of type Φ over R and its elementary subgroup . This paper investigates the normalizers of and in the larger group , where S is an extension ring of R. We establish that under certain conditions on R these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to .
{"title":"Normalizer of twisted Chevalley groups over commutative rings","authors":"Shripad M. Garge , Deep H. Makadiya","doi":"10.1016/j.jpaa.2025.108094","DOIUrl":"10.1016/j.jpaa.2025.108094","url":null,"abstract":"<div><div>Let <em>R</em> be a commutative ring with unity. Consider the twisted Chevalley group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> of type Φ over <em>R</em> and its elementary subgroup <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span>. This paper investigates the normalizers of <span><math><msubsup><mrow><mi>E</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow><mrow><mo>′</mo></mrow></msubsup><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span> in the larger group <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span>, where <em>S</em> is an extension ring of <em>R</em>. We establish that under certain conditions on <em>R</em> these normalizers coincide. Moreover, in the case of adjoint type groups, we show that they are precisely equal to <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>π</mi><mo>,</mo><mi>σ</mi></mrow></msub><mo>(</mo><mi>Φ</mi><mo>,</mo><mi>R</mi><mo>)</mo></math></span>.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108094"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145109501","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-11-01Epub Date: 2025-10-20DOI: 10.1016/j.jpaa.2025.108112
Neil Epstein
The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be fundamental, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of local fundamentality is introduced and its behavior developed. On the algebraic side, the notions of locally, geometrically, and generically unit-additive rings are introduced, thus complementing unit-additivity as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.
{"title":"Fundamental algebraic sets and locally unit-additive rings","authors":"Neil Epstein","doi":"10.1016/j.jpaa.2025.108112","DOIUrl":"10.1016/j.jpaa.2025.108112","url":null,"abstract":"<div><div>The Fundamental Theorem of Algebra can be thought of as a statement about the real numbers as a space, considered as an algebraic set over the real numbers as a field. This paper introduces what it means for an algebraic set or affine variety over a field to be <em>fundamental</em>, in a way that encompasses the Fundamental Theorem of Algebra as a special case. The related concept of <em>local</em> fundamentality is introduced and its behavior developed. On the algebraic side, the notions of <em>locally</em>, <em>geometrically</em>, and <em>generically unit-additive</em> rings are introduced, thus complementing <em>unit-additivity</em> as previously defined by the author and Jay Shapiro. A number of results are extended from the previous joint paper from unit-additivity to local unit-additivity. It is shown that an affine variety is (locally) fundamental if and only if its coordinate ring is (locally) unit-additive. To do so, a theorem is proved showing that there are many equivalent definitions of local unit-additivity. Illustrative examples are sprinkled throughout.</div></div>","PeriodicalId":54770,"journal":{"name":"Journal of Pure and Applied Algebra","volume":"229 11","pages":"Article 108112"},"PeriodicalIF":0.8,"publicationDate":"2025-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145416440","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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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