International Journal of Algebra and Computation最新文献

{"title":"Clonoids between modules","authors":"Peter Mayr, Patrick Wynne","doi":"10.1142/s021819672450022x","DOIUrl":"https://doi.org/10.1142/s021819672450022x","url":null,"abstract":"<p>Clonoids are sets of finitary functions from an algebra <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> to an algebra <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span> that are closed under composition with term functions of <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> on the domain side and with term functions of <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span> on the codomain side. For <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A, B</mtext></mstyle></math></span><span></span> (polynomially equivalent to) finite modules we show: If <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A, B</mtext></mstyle></math></span><span></span> have coprime order and the congruence lattice of <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> is distributive, then there are only finitely many clonoids from <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> to <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span>. This is proved by establishing for every natural number <span><math altimg=\"eq-00010.gif\" display=\"inline\"><mi>k</mi></math></span><span></span> a particular linear equation that all <span><math altimg=\"eq-00011.gif\" display=\"inline\"><mi>k</mi></math></span><span></span>-ary functions from <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> to <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span> satisfy. Else if <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A, B</mtext></mstyle></math></span><span></span> do not have coprime order, then there exist infinite ascending chains of clonoids from <span><math altimg=\"eq-00015.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> to <span><math altimg=\"eq-00016.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span> ordered by inclusion. Consequently any extension of <span><math altimg=\"eq-00017.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">A</mtext></mstyle></math></span><span></span> by <span><math altimg=\"eq-00018.gif\" display=\"inline\"><mstyle><mtext mathvariant=\"normal\">B</mtext></mstyle></math></span><span></span> has countably infinitely many <","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"36 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141198021","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}

{"title":"There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups","authors":"Mikhail Anokhin","doi":"10.1142/s0218196724500188","DOIUrl":"https://doi.org/10.1142/s0218196724500188","url":null,"abstract":"<p>Let <span><math altimg=\"eq-00001.gif\" display=\"inline\"><mi mathvariant=\"normal\">Ω</mi></math></span><span></span> be a finite set of finitary operation symbols and let <span><math altimg=\"eq-00002.gif\" display=\"inline\"><mi>𝔙</mi></math></span><span></span> be a nontrivial variety of <span><math altimg=\"eq-00003.gif\" display=\"inline\"><mi mathvariant=\"normal\">Ω</mi></math></span><span></span>-algebras. Assume that for some set <span><math altimg=\"eq-00004.gif\" display=\"inline\"><mi mathvariant=\"normal\">Γ</mi><mo>⊆</mo><mi mathvariant=\"normal\">Ω</mi></math></span><span></span> of group operation symbols, all <span><math altimg=\"eq-00005.gif\" display=\"inline\"><mi mathvariant=\"normal\">Ω</mi></math></span><span></span>-algebras in <span><math altimg=\"eq-00006.gif\" display=\"inline\"><mi>𝔙</mi></math></span><span></span> are groups under the operations associated with the symbols in <span><math altimg=\"eq-00007.gif\" display=\"inline\"><mi mathvariant=\"normal\">Γ</mi></math></span><span></span>. In other words, <span><math altimg=\"eq-00008.gif\" display=\"inline\"><mi>𝔙</mi></math></span><span></span> is assumed to be a nontrivial variety of expanded groups. In particular, <span><math altimg=\"eq-00009.gif\" display=\"inline\"><mi>𝔙</mi></math></span><span></span> can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in <span><math altimg=\"eq-00010.gif\" display=\"inline\"><mi>𝔙</mi></math></span><span></span>, even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families <span><math altimg=\"eq-00011.gif\" display=\"inline\"><mo stretchy=\"false\">(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msub><mi>|d</mi><mo>∈</mo><mi>D</mi><mo stretchy=\"false\">)</mo></math></span><span></span> of computational and black-box <span><math altimg=\"eq-00012.gif\" display=\"inline\"><mi mathvariant=\"normal\">Ω</mi></math></span><span></span>-algebras (where <span><math altimg=\"eq-00013.gif\" display=\"inline\"><mi>D</mi><mo>⊆</mo><msup><mrow><mo stretchy=\"false\">{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo stretchy=\"false\">}</mo></mrow><mrow><mo stretchy=\"false\">∗</mo></mrow></msup></math></span><span></span>) such that for every <span><math altimg=\"eq-00014.gif\" display=\"inline\"><mi>d</mi><mo>∈</mo><mi>D</mi></math></span><span></span>, each element of <span><math altimg=\"eq-00015.gif\" display=\"inline\"><msub><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msub></math></span><span></span> is represented by a unique bit string of length polynomial in the length of <span><math altimg=\"eq-00016.gif\" display=\"inline\"><mi>d</mi></math></span><span></span>. In our main result, we use straight-line programs to represent nontrivial relations between elements of <span><math altimg=\"eq-00017.gif\" display=\"inline\"><mi mathvariant=\"normal\">Ω</mi></math></span><span></span>-algebras. Note that under certain conditions, this result depends on the classification of finite simple gr","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"2010 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190865","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}

{"title":"On the lattice of closed subgroups of a profinite group","authors":"Francesco de Giovanni, Iker de las Heras, Marco Trombetti","doi":"10.1142/s0218196724500206","DOIUrl":"https://doi.org/10.1142/s0218196724500206","url":null,"abstract":"<p>The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behavior of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"84 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140926562","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}

{"title":"An algorithm to recognize echelon subgroups of a free group","authors":"Dario Ascari","doi":"10.1142/s021819672450019x","DOIUrl":"https://doi.org/10.1142/s021819672450019x","url":null,"abstract":"<p>We provide an algorithm that, given a finite set of generators for a subgroup <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> of a finitely generated free group <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>F</mi></math></span><span></span>, determines whether <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> is echelon or not and, in case of affirmative answer, also computes a basis with respect to which <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>H</mi></math></span><span></span> is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"12 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140841938","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}

{"title":"Properties of congruence lattices of graph inverse semigroups","authors":"Marina Anagnostopoulou-Merkouri, Zachary Mesyan, James D. Mitchell","doi":"10.1142/s0218196724500139","DOIUrl":"https://doi.org/10.1142/s0218196724500139","url":null,"abstract":"<p>From any directed graph <span><math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span> one can construct the graph inverse semigroup <span><math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo></math></span><span></span>, whose elements, roughly speaking, correspond to paths in <span><math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span>. Wang and Luo showed that the congruence lattice <span><math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> of <span><math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo></math></span><span></span> is upper-semimodular for every graph <span><math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span>, but can fail to be lower-semimodular for some <span><math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span>. We provide a simple characterization of the graphs <span><math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span> for which <span><math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> is lower-semimodular. We also describe those <span><math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span> such that <span><math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> is atomistic, and characterize the minimal generating sets for <span><math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"><mi>L</mi><mo stretchy=\"false\">(</mo><mi>G</mi><mo stretchy=\"false\">(</mo><mi>E</mi><mo stretchy=\"false\">)</mo><mo stretchy=\"false\">)</mo></math></span><span></span> when <span><math altimg=\"eq-00013.gif\" display=\"inline\" overflow=\"scroll\"><mi>E</mi></math></span><span></span> is finite and simple.</p>","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"15 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2024-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806223","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}

{"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":"&lt;p&gt;We investigate the power of counting in &lt;span&gt;Group Isomorphism&lt;/span&gt;. 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 &lt;i&gt;35th Annual ACM/IEEE Symp. Logic in Computer Scienc&lt;/i&gt;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:&lt;/p&gt;&lt;ul&gt;&lt;li&gt;&lt;p&gt;Direct products of non-Abelian simple groups.&lt;/p&gt;&lt;/li&gt;&lt;li&gt;&lt;p&gt;Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-generated solvable group with solvability class poly log log &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. This notably includes instances where the complement is an &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;-generated nilpotent group. This problem was previously known to be in &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"sans-serif\"&gt;P&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; [Y. Qiao, J. M. N. Sarma and B. Tang, On isomorphism testing of groups with normal Hall subgroups, in &lt;i&gt;Proc. 28th Symp. Theoretical Aspects of Computer Science,&lt;/i&gt; Dagstuhl Castle, Leibniz Center for Informatics, 2011), pp. 567–578, doi:10.4230/LIPIcs. STACS.2011.567], and the complexity was recently improved to &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"sans-serif\"&gt;L&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; [J. A. Grochow and M. Levet, On the parallel complexity of group isomorphism via Weisfeiler–Leman, in &lt;i&gt;24th Int. Symp. Fundamentals of Computation Theory&lt;/i&gt;, eds. H. Fernau and K. Jansen, Lecture Notes in Computer Science, Vol. 14292, September 18–21, 2023, Trier, Germany (Springer, 2023), pp. 234–247].&lt;/p&gt;&lt;/li&gt;&lt;li&gt;&lt;p&gt;Graphical groups of class &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and exponent &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mo&gt;&gt;&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; [A. H. Mekler, Stability of nilpotent groups of class 2 and prime exponent, &lt;i&gt;J. Symb. Logic&lt;/i&gt;&lt;b&gt;46&lt;/b&gt;(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, &lt;i&gt;Combinatorica&lt;/i&gt;&lt;b&gt;12&lt;/b&gt;(4) (1992) 389–410], respectively. In particular, our work imp","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"47 1","pages":""},"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}

{"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":"243 1","pages":""},"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}

{"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":"&lt;p&gt;The Hilbert scheme &lt;span&gt;&lt;math altimg=\"eq-00001.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Hilb&lt;/mtext&gt;&lt;/mstyle&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; of &lt;span&gt;&lt;math altimg=\"eq-00002.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; points on &lt;span&gt;&lt;math altimg=\"eq-00003.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; can be expressed as the critical locus of a regular function on a smooth variety &lt;span&gt;&lt;math altimg=\"eq-00004.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi mathvariant=\"cal\"&gt;𝒳&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. Recent development in birational geometry suggests a study of singularities of the pair &lt;span&gt;&lt;math altimg=\"eq-00005.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi mathvariant=\"cal\"&gt;𝒳&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Hilb&lt;/mtext&gt;&lt;/mstyle&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; using jet schemes. In this paper, we use a comparison between &lt;span&gt;&lt;math altimg=\"eq-00006.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Hilb&lt;/mtext&gt;&lt;/mstyle&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and the scheme &lt;span&gt;&lt;math altimg=\"eq-00007.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;msub&gt;&lt;mrow&gt;&lt;mi&gt;C&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; of three commuting &lt;span&gt;&lt;math altimg=\"eq-00008.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;×&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; matrices to estimate the log canonical threshold of &lt;span&gt;&lt;math altimg=\"eq-00009.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;mi mathvariant=\"cal\"&gt;𝒳&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Hilb&lt;/mtext&gt;&lt;/mstyle&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;. As a consequence, we see that although both &lt;span&gt;&lt;math altimg=\"eq-00010.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;mi mathvariant=\"cal\"&gt;𝒳&lt;/mi&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; and &lt;span&gt;&lt;math altimg=\"eq-00011.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mo&gt;dim&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mstyle&gt;&lt;mtext mathvariant=\"normal\"&gt;Hilb&lt;/mtext&gt;&lt;/mstyle&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;ℂ&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt; have asymptotic growth &lt;span&gt;&lt;math altimg=\"eq-00012.gif\" display=\"inline\" overflow=\"scroll\"&gt;&lt;mi&gt;O&lt;/mi&gt;&lt;mo stretchy=\"false\"&gt;(&lt;/mo&gt;&lt;msup&gt;&lt;mrow&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/mrow&gt;&lt;mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mo stretchy=\"false\"&gt;)&lt;/mo&gt;&lt;/math&gt;&lt;/span&gt;&lt;span&gt;&lt;/span&gt;,","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"245 1","pages":""},"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}

{"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":"54 1","pages":""},"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}

{"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":"20 1","pages":""},"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}