Pub Date : 2024-04-03DOI: 10.1142/s0218196724500103
Nathaniel A. Collins, Michael Levet
We investigate the power of counting in Group Isomorphism. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp. Logic in Computer Science, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include:
Direct products of non-Abelian simple groups.
Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an -generated solvable group with solvability class poly log log . This notably includes instances where the complement is an -generated nilpotent group. This problem was previously known to be in [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc. 28th Symp. Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in 24th Int. Symp. Fundamentals of Computation Theory, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247].
Graphical groups of class and exponent [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb. Logic46(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J. -Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica12(4) (1992) 389–410], respectively. In particular, our work imp
我们研究了群同构中计数的力量。我们首先利用群的 Weisfeiler-Leman 第一版算法的无计数变体 [J. Brachter and P. Schweitzer, On Weisfeiler-Leman dimension of finite groups, in 35 Annual ACM/IEEE Symposium.Brachter and P. Schweitzer, On the Weisfeiler-Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp.Logic in Computer Science, eds.H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8-11, 2020 (ACM, 2020), pp.这些群族包括:非阿贝尔简单群的直接乘积;正态霍尔子群是阿贝尔群,而补集是可解类为 poly log log n 的 O(1)- 生成的可解群。这个问题之前已知在 P [Y. Qiao, J. M. N.] 中。Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in Proc.Theoretical Aspects of Computer Science, Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp.STACS.2011.567],最近又将复杂度提高到了 L [J. A. Grochow and M. M. J. J. M. M.A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler-Leman, in 24th Int.Symp.计算理论基础》,H. Fernau 和 K. Levet 编辑。H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18-21, Trier, Germany (Springer, 2023), pp.H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, J. Symb.Logic46(4) (1981) 781-788] 由 CFI 和扭曲 CFI 图产生 [J. -Y.Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, Combinatorica12(4) (1992) 389-410], respectively.尤其是,我们的工作改进了 Brachter 和 Schweitzer [On the Weisfeiler-Leman dimension of finite groups, in 35th Annual ACM/IEEE Symp.Logic in Computer Science, eds.H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8-11, 2020 (ACM, 2020), pp.最后,我们证明了 qary 无计数卵石博弈甚至无法区分阿贝尔群。这扩展了格罗霍和勒维特(Grochow and Levet)(同上)的结果,他们是在 q=1 的情况下建立这一结果的。总的主题是,要把群同构放到 P 中,似乎需要一些计数。
{"title":"Count-free Weisfeiler–Leman and group isomorphism","authors":"Nathaniel A. Collins, Michael Levet","doi":"10.1142/s0218196724500103","DOIUrl":"https://doi.org/10.1142/s0218196724500103","url":null,"abstract":"<p>We investigate the power of counting in <span>Group Isomorphism</span>. We first leverage the count-free variant of the Weisfeiler–Leman Version I algorithm for groups [J. Brachter and P. Schweitzer, On the Weisfeiler–Leman dimension of finite groups, in <i>35th Annual ACM/IEEE Symp. Logic in Computer Scienc</i>e, eds. H. Hermanns, L. Zhang, N. Kobayashi and D. Miller, Saarbrucken, Germany, July 8–11, 2020 (ACM, 2020), pp. 287–300, doi:10.1145/3373718.3394786] in tandem with bounded non-determinism and limited counting to improve the parallel complexity of isomorphism testing for several families of groups. These families include:</p><ul><li><p>Direct products of non-Abelian simple groups.</p></li><li><p>Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>-generated solvable group with solvability class poly log log <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span>. This notably includes instances where the complement is an <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><mn>1</mn><mo stretchy=\"false\">)</mo></math></span><span></span>-generated nilpotent group. This problem was previously known to be in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"sans-serif\">P</mi></math></span><span></span> [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in <i>Proc. 28th Symp. Theoretical Aspects of Computer Science,</i> Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"sans-serif\">L</mi></math></span><span></span> [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in <i>24th Int. Symp. Fundamentals of Computation Theory</i>, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247].</p></li><li><p>Graphical groups of class <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mn>2</mn></math></span><span></span> and exponent <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>p</mi><mo>></mo><mn>2</mn></math></span><span></span> [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, <i>J. Symb. Logic</i><b>46</b>(4) (1981) 781–788] arising from the CFI and twisted CFI graphs [J. -Y. Cai, M. Fürer and N. Immerman, An optimal lower bound on the number of variables for graph identification, <i>Combinatorica</i><b>12</b>(4) (1992) 389–410], respectively. In particular, our work imp","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563344","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 : 2024-04-03DOI: 10.1142/s0218196724500085
Gilberto B. Almeida Filho, Matheus Bernardini
We bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to m-extensions. It allows us to identify gapsets and, in general, m-extensions with tilings of boards; as a consequence, we present some applications of this identification. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multiplicity is 3 or 4, and, in some cases, for the number of gapsets with fixed genus and depth.
{"title":"Gapsets and the k-generalized Fibonacci sequences","authors":"Gilberto B. Almeida Filho, Matheus Bernardini","doi":"10.1142/s0218196724500085","DOIUrl":"https://doi.org/10.1142/s0218196724500085","url":null,"abstract":"<p>We bring the terminology of the Kunz coordinates of numerical semigroups to gapsets and we generalize this concept to <i>m</i>-extensions. It allows us to identify gapsets and, in general, <i>m</i>-extensions with tilings of boards; as a consequence, we present some applications of this identification. Moreover, we present explicit formulas for the number of gapsets with fixed genus and depth, when the multiplicity is 3 or 4, and, in some cases, for the number of gapsets with fixed genus and depth.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597591","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 : 2024-04-03DOI: 10.1142/s0218196724500140
Jen-Chieh Hsiao
The Hilbert scheme of points on can be expressed as the critical locus of a regular function on a smooth variety . Recent development in birational geometry suggests a study of singularities of the pair using jet schemes. In this paper, we use a comparison between and the scheme of three commuting matrices to estimate the log canonical threshold of . As a consequence, we see that although both and have asymptotic growth ,
{"title":"Bounding embedded singularities of Hilbert schemes of points on affine three space","authors":"Jen-Chieh Hsiao","doi":"10.1142/s0218196724500140","DOIUrl":"https://doi.org/10.1142/s0218196724500140","url":null,"abstract":"<p>The Hilbert scheme <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> of <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi></math></span><span></span> points on <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> can be expressed as the critical locus of a regular function on a smooth variety <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi mathvariant=\"cal\">𝒳</mi></math></span><span></span>. Recent development in birational geometry suggests a study of singularities of the pair <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒳</mi><mo>,</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span> using jet schemes. In this paper, we use a comparison between <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> and the scheme <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>C</mi></mrow><mrow><mn>3</mn><mo>,</mo><mi>n</mi></mrow></msub></math></span><span></span> of three commuting <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo stretchy=\"false\">×</mo><mi>n</mi></math></span><span></span> matrices to estimate the log canonical threshold of <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi mathvariant=\"cal\">𝒳</mi><mo>,</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span>. As a consequence, we see that although both <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mo>dim</mo><mi mathvariant=\"cal\">𝒳</mi></math></span><span></span> and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mo>dim</mo><msup><mrow><mstyle><mtext mathvariant=\"normal\">Hilb</mtext></mstyle></mrow><mrow><mi>n</mi></mrow></msup><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span><span></span> have asymptotic growth <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>O</mi><mo stretchy=\"false\">(</mo><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy=\"false\">)</mo></math></span><span></span>,","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140563786","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 : 2024-04-03DOI: 10.1142/s0218196724500097
C. E. Kofinas, A. I. Papistas
For a positive integer , let be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank n. We show that the subgroup of generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of is dense in with respect to the formal power series topology.
{"title":"On automorphisms of certain free nilpotent-by-abelian Lie algebras","authors":"C. E. Kofinas, A. I. Papistas","doi":"10.1142/s0218196724500097","DOIUrl":"https://doi.org/10.1142/s0218196724500097","url":null,"abstract":"<p>For a positive integer <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>n</mi><mo>≥</mo><mn>4</mn></math></span><span></span>, let <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> be a free (nilpotent of class 2)-by-abelian and abelian-by-(nilpotent of class 2) Lie algebra of rank <i>n</i>. We show that the subgroup of <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Aut</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span><span></span> is dense in <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">Aut</mtext></mstyle><mo stretchy=\"false\">(</mo><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub><mo stretchy=\"false\">)</mo></math></span><span></span> with respect to the formal power series topology.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140597595","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 : 2024-03-26DOI: 10.1142/s0218196724500115
Peter V. Danchev, Truong Huu Dung, Tran Nam Son
In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.
{"title":"Products of traceless and semi-traceless matrices over division rings and their applications","authors":"Peter V. Danchev, Truong Huu Dung, Tran Nam Son","doi":"10.1142/s0218196724500115","DOIUrl":"https://doi.org/10.1142/s0218196724500115","url":null,"abstract":"<p>In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140315633","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 : 2024-03-15DOI: 10.1142/s0218196724500152
Vítězslav Kala, T. Kepka, M. Korbelář
{"title":"Congruence-simple matrix semirings","authors":"Vítězslav Kala, T. Kepka, M. Korbelář","doi":"10.1142/s0218196724500152","DOIUrl":"https://doi.org/10.1142/s0218196724500152","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140238183","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 : 2024-03-07DOI: 10.1142/s0218196724500012
Priyavrat Deshpande, Mallika Roy
Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right-angled Artin groups can be expressed in terms of their defining graph.
In this paper, we address the question of understanding the structure of a class of subgroups of right-angled Artin groups in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina–Brady groups or Artin kernels) from a finiteness conditions viewpoint. Unlike the right-angled Artin groups the isomorphism type of Bestvina–Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina–Brady groups can be expressed as an iterated amalgamated product. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient right-angled Artin group.
{"title":"On the structure of finitely presented Bestvina–Brady groups","authors":"Priyavrat Deshpande, Mallika Roy","doi":"10.1142/s0218196724500012","DOIUrl":"https://doi.org/10.1142/s0218196724500012","url":null,"abstract":"<p>Right-angled Artin groups and their subgroups are of great interest because of their geometric, combinatorial and algorithmic properties. It is convenient to define these groups using finite simplicial graphs. The isomorphism type of the group is uniquely determined by the graph. Moreover, many structural properties of right-angled Artin groups can be expressed in terms of their defining graph.</p><p>In this paper, we address the question of understanding the structure of a class of subgroups of right-angled Artin groups in terms of the graph. Bestvina and Brady, in their seminal work, studied these subgroups (now called Bestvina–Brady groups or Artin kernels) from a finiteness conditions viewpoint. Unlike the right-angled Artin groups the isomorphism type of Bestvina–Brady groups is not uniquely determined by the defining graph. We prove that certain finitely presented Bestvina–Brady groups can be expressed as an iterated amalgamated product. Moreover, we show that this amalgamated product can be read off from the graph defining the ambient right-angled Artin group.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140105993","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 : 2024-03-07DOI: 10.1142/s0218196724500024
Jana Volaříková
We deal with the question of the -reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids is called -reducible if, given a finite ordered monoid M, for every inequality of pseudowords that is valid in , there exists an inequality of -words that is also valid in and has the same “imprint” in M.
Place and Zeitoun have recently proven the decidability of the membership problem for levels , 1, and of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels and are -reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels
我们要讨论的问题是与正则表达式语言的连接层次相对应的有序单元的伪变体的ω-可还原性。如果给定一个有限有序单元 M,对于在 V 中有效的每一个伪词不等式,都存在一个在 V 中也有效并且在 M 中具有相同 "印记 "的 ω 词不等式,那么有序单元的伪变量 V 就被称为 ω 可还原性。Place 和 Zeitoun 最近证明了第 1∕2、1、3∕2 和 5∕2 层连接层次的成员资格问题的可解性,其中第 0 层是在商下封闭的正则表达式语言的有限布尔代数。这些成员问题的解法是通过考虑更一般的正则表达式语言分离问题及其进一步推广--覆盖问题--而找到的。根据 Place 和 Zeitoun 的研究成果,我们证明了,对于每个由局部有限单体伪变体表示第 0 层的连接层次,对应于第 1∕2 层和第 3∕2 层的伪变体都是ω-可还原的。作为这些结果的推论,我们得到,对于每一个层次 0 由局部有限的单体伪变体表示的并集层次,对应于层次 3∕2 和 5∕2 的伪变体都可以用 ω-inequalities 来定义。此外,在 Straubing-Thérien 层次结构的特殊情况下,利用 Place 和 Zeitoun 提出的层次 2 特性定理,我们可以得到层次 2 是可以用 ω-identity 来定义的。
{"title":"The omega-reducibility of pseudovarieties of ordered monoids representing low levels of concatenation hierarchies","authors":"Jana Volaříková","doi":"10.1142/s0218196724500024","DOIUrl":"https://doi.org/10.1142/s0218196724500024","url":null,"abstract":"<p>We deal with the question of the <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducibility of pseudovarieties of ordered monoids corresponding to levels of concatenation hierarchies of regular languages. A pseudovariety of ordered monoids <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span> is called <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducible if, given a finite ordered monoid <i>M</i>, for every inequality of pseudowords that is valid in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span>, there exists an inequality of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-words that is also valid in <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mstyle mathvariant=\"sans-serif\"><mi>V</mi></mstyle></math></span><span></span> and has the same “imprint” in <i>M</i>.</p><p>Place and Zeitoun have recently proven the decidability of the membership problem for levels <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span>, 1, <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mn>5</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> of concatenation hierarchies with level 0 being a finite Boolean algebra of regular languages closed under quotients. The solutions of these membership problems have been found by considering a more general problem of separation of regular languages and its further generalization — a problem of covering. Following the results of Place and Zeitoun, we prove that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mn>1</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> and <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span> are <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>ω</mi></math></span><span></span>-reducible. As a corollary of these results, we obtain that, for every concatenation hierarchy with level 0 being represented by a locally finite pseudovariety of monoids, the pseudovarieties corresponding to levels <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mn>3</mn><mo stretchy=\"false\">∕</mo><mn>2</mn></math></span><span></span","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140106337","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 : 2024-02-20DOI: 10.1142/s0218196724500061
Pavel Shumyatsky, Matteo Vannacci
Let be the probability that two random elements of a finite ring commute and the probability that the product of two random elements in is zero. We show that if , then there exists a Lie-ideal in the Lie-ring with -bounded index and with of -bounded order. If , then there exists an ideal in with -bounded index and of -bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.
设 cp(R) 是有限环 R 中两个随机元素相通的概率,zp(R) 是 R 中两个随机元素的乘积为零的概率。我们将证明,如果 cp(R)=𝜀 ,那么在李环(R,[⋅,⋅])中存在一个具有𝜀 边界索引和具有𝜀 边界阶的 [D,D] 的李偶像 D。如果 zp(R)=𝜀, 那么 R 中存在一个索引为𝜀 的理想 D,且 D2 的阶为𝜀。这些结果类似于诺伊曼关于有限群中交换概率的著名定理。
{"title":"Commuting and product-zero probability in finite rings","authors":"Pavel Shumyatsky, Matteo Vannacci","doi":"10.1142/s0218196724500061","DOIUrl":"https://doi.org/10.1142/s0218196724500061","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span> be the probability that two random elements of a finite ring <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> commute and <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">zp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo></math></span><span></span> the probability that the product of two random elements in <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> is zero. We show that if <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">cp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>𝜀</mi></math></span><span></span>, then there exists a Lie-ideal <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi></math></span><span></span> in the Lie-ring <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">(</mo><mi>R</mi><mo>,</mo><mo stretchy=\"false\">[</mo><mo stretchy=\"false\">⋅</mo><mo>,</mo><mo stretchy=\"false\">⋅</mo><mo stretchy=\"false\">]</mo><mo stretchy=\"false\">)</mo></math></span><span></span> with <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded index and with <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mo stretchy=\"false\">[</mo><mi>D</mi><mo>,</mo><mi>D</mi><mo stretchy=\"false\">]</mo></math></span><span></span> of <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded order. If <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mstyle><mtext mathvariant=\"normal\">zp</mtext></mstyle><mo stretchy=\"false\">(</mo><mi>R</mi><mo stretchy=\"false\">)</mo><mo>=</mo><mi>𝜀</mi></math></span><span></span>, then there exists an ideal <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>D</mi></math></span><span></span> in <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>R</mi></math></span><span></span> with <span><math altimg=\"eq-00014.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded index and <span><math altimg=\"eq-00015.gif\" display=\"inline\" overflow=\"scroll\"><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span><span></span> of <span><math altimg=\"eq-00016.gif\" display=\"inline\" overflow=\"scroll\"><mi>𝜀</mi></math></span><span></span>-bounded order. These results are analogous to the well-known theorem of Neumann on the commuting probability in finite groups.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045292","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 : 2024-01-31DOI: 10.1142/s021819672450005x
Rongmin Zhu, Haiyu Liu
{"title":"Generalized Gorenstein modules with respect to duality pairs over triangular matrix rings","authors":"Rongmin Zhu, Haiyu Liu","doi":"10.1142/s021819672450005x","DOIUrl":"https://doi.org/10.1142/s021819672450005x","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140470909","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}