International Journal of Algebra and Computation最新文献

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IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-05-25 DOI: 10.1142/s021819672450022x

Peter Mayr, Patrick Wynne

Clonoids are sets of finitary functions from an algebra <math altimg="eq-00001.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> to an algebra <math altimg="eq-00002.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math> that are closed under composition with term functions of <math altimg="eq-00003.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> on the domain side and with term functions of <math altimg="eq-00004.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math> on the codomain side. For <math altimg="eq-00005.gif" display="inline"><mstyle><mtext mathvariant="normal">A, B</mtext></mstyle></math> (polynomially equivalent to) finite modules we show: If <math altimg="eq-00006.gif" display="inline"><mstyle><mtext mathvariant="normal">A, B</mtext></mstyle></math> have coprime order and the congruence lattice of <math altimg="eq-00007.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> is distributive, then there are only finitely many clonoids from <math altimg="eq-00008.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> to <math altimg="eq-00009.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math>. This is proved by establishing for every natural number <math altimg="eq-00010.gif" display="inline"><mi>k</mi></math> a particular linear equation that all <math altimg="eq-00011.gif" display="inline"><mi>k</mi></math>-ary functions from <math altimg="eq-00012.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> to <math altimg="eq-00013.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math> satisfy. Else if <math altimg="eq-00014.gif" display="inline"><mstyle><mtext mathvariant="normal">A, B</mtext></mstyle></math> do not have coprime order, then there exist infinite ascending chains of clonoids from <math altimg="eq-00015.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> to <math altimg="eq-00016.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math> ordered by inclusion. Consequently any extension of <math altimg="eq-00017.gif" display="inline"><mstyle><mtext mathvariant="normal">A</mtext></mstyle></math> by <math altimg="eq-00018.gif" display="inline"><mstyle><mtext mathvariant="normal">B</mtext></mstyle></math> has countably infinitely many <

有限模块是指从代数 A 到代数 B 的有限函数集合，这些函数在与域边上的 A 的项函数和同域边上的 B 的项函数的组合下是封闭的。对于 A、B（多项式等价于）有限模块，我们证明：如果 A、B 有共阶，且 A 的全等网格是分布式的，那么从 A 到 B 只有有限多个克隆子。这可以通过为每个自然数 k 建立一个特定的线性方程来证明，从 A 到 B 的所有 kary 函数都满足这个方程。否则，如果 A、B 没有共序，那么就存在从 A 到 B 按包含排序的无限递增的克隆子链。因此，任何由 B 扩展的 A 都有可数的无限多个 2-nilpotent 扩展，直到项等价。

引用次数: 0

There are no post-quantum weakly pseudo-free families in any nontrivial variety of expanded groups 在任何非数量级的扩展群中都不存在后量子弱伪自由族

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-05-17 DOI: 10.1142/s0218196724500188

Mikhail Anokhin

Let <math altimg="eq-00001.gif" display="inline"><mi mathvariant="normal">Ω</mi></math> be a finite set of finitary operation symbols and let <math altimg="eq-00002.gif" display="inline"><mi>𝔙</mi></math> be a nontrivial variety of <math altimg="eq-00003.gif" display="inline"><mi mathvariant="normal">Ω</mi></math>-algebras. Assume that for some set <math altimg="eq-00004.gif" display="inline"><mi mathvariant="normal">Γ</mi><mo>⊆</mo><mi mathvariant="normal">Ω</mi></math> of group operation symbols, all <math altimg="eq-00005.gif" display="inline"><mi mathvariant="normal">Ω</mi></math>-algebras in <math altimg="eq-00006.gif" display="inline"><mi>𝔙</mi></math> are groups under the operations associated with the symbols in <math altimg="eq-00007.gif" display="inline"><mi mathvariant="normal">Γ</mi></math>. In other words, <math altimg="eq-00008.gif" display="inline"><mi>𝔙</mi></math> is assumed to be a nontrivial variety of expanded groups. In particular, <math altimg="eq-00009.gif" display="inline"><mi>𝔙</mi></math> can be a nontrivial variety of groups or rings. Our main result is that there are no post-quantum weakly pseudo-free families in <math altimg="eq-00010.gif" display="inline"><mi>𝔙</mi></math>, even in the worst-case setting and/or the black-box model. In this paper, we restrict ourselves to families <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> of computational and black-box <math altimg="eq-00012.gif" display="inline"><mi mathvariant="normal">Ω</mi></math>-algebras (where <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>) such that for every <math altimg="eq-00014.gif" display="inline"><mi>d</mi><mo>∈</mo><mi>D</mi></math>, each element of <math altimg="eq-00015.gif" display="inline"><msub><mrow><mi>H</mi></mrow><mrow><mi>d</mi></mrow></msub></math> is represented by a unique bit string of length polynomial in the length of <math altimg="eq-00016.gif" display="inline"><mi>d</mi></math>. In our main result, we use straight-line programs to represent nontrivial relations between elements of <math altimg="eq-00017.gif" display="inline"><mi mathvariant="normal">Ω</mi></math>-algebras. Note that under certain conditions, this result depends on the classification of finite simple gr

让 Ω 是有限运算符号集，让 𝔙 是 Ω-gebras 的非奇异集合。假设对于某个群运算符号集 Γ⊆Ω，𝔙 中的所有 Ω-gebras 在与Γ 中符号相关的运算下都是群。换句话说，我们假定𝔙 是一个扩展群的非小类。特别是，𝔙 可以是群或环的一个非私密种类。我们的主要结果是，即使在最坏情况设置和/或黑箱模型中，𝔙中也不存在后量子弱无伪族。在本文中，我们将自己限制在计算和黑箱Ω-数组（其中 D⊆{0,1}∗）的族 (Hd|d∈D)，这样对于每个 d∈D，Hd 的每个元素都由长度与 d 的长度成多项式的唯一比特串来表示。请注意，在某些条件下，这一结果取决于有限简单群的分类。此外，我们还定义并研究了计算和黑盒子Ω玻家族的一些后量子弱伪无穷性类型。

引用次数: 0

On the lattice of closed subgroups of a profinite group 论无限群的封闭子群网格

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-05-09 DOI: 10.1142/s0218196724500206

Francesco de Giovanni, Iker de las Heras, Marco Trombetti

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.

一个群的子群网格是有关群本身结构的重要信息来源。本文旨在使用类似的工具来研究无限群。我们将更详细地研究无限群的封闭子群或开放子群的网格及其与整个群的关系。例如，我们证明了原环群是唯一具有封闭子群或开放子群的分布晶格的无限群，并给出了封闭（或开放）子群晶格满足戴德金模态律的无限群的尖锐特征；实际上，我们给出了封闭子群晶格的模态元素行为的精确描述。我们还处理了从一个无限群到另一个具有同构封闭（或开放）子群网格的无限群之间传递某些结构信息的问题。此外，我们还得到了一些有趣的结果以及与可分解性和具有给定封闭（或开放）子群网格的无限群数量有关的相关结果。

引用次数: 0

An algorithm to recognize echelon subgroups of a free group 识别自由群梯形子群的算法

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-30 DOI: 10.1142/s021819672450019x

Dario Ascari

We provide an algorithm that, given a finite set of generators for a subgroup $H$ of a finitely generated free group $F$ , determines whether $H$ is echelon or not and, in case of affirmative answer, also computes a basis with respect to which $H$ 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.

我们提供了一种算法，只要给定有限生成的自由群 F 的子群 H 的有限生成子集，就能确定 H 是否为梯形，如果答案是肯定的，还能计算出 H 为梯形的基。这就回答了罗森曼的一个问题。我们还通过一个反例证明了两个梯形子群的交集不一定是梯形，从而回答了罗森曼的另一个问题。

引用次数: 0

Properties of congruence lattices of graph inverse semigroups 图逆半群全等网格的性质

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-20 DOI: 10.1142/s0218196724500139

Marina Anagnostopoulou-Merkouri, Zachary Mesyan, James D. Mitchell

From any directed graph $E$ one can construct the graph inverse semigroup $G (E)$ , whose elements, roughly speaking, correspond to paths in $E$ . Wang and Luo showed that the congruence lattice $L (G (E))$ of $G (E)$ is upper-semimodular for every graph $E$ , but can fail to be lower-semimodular for some $E$ . We provide a simple characterization of the graphs $E$ for which $L (G (E))$ is lower-semimodular. We also describe those $E$ such that $L (G (E))$ is atomistic, and characterize the minimal generating sets for $L (G (E))$ when $E$ is finite and simple.

王和罗（Wang and Luo）的研究表明，G(E) 的同余网格 L(G(E)) 对于每个图 E 都是上半模的，但对于某些图 E 可能不是下半模的。我们还描述了那些 L(G(E)) 原子化的 E，并描述了当 E 有限且简单时 L(G(E)) 的最小生成集。

引用次数: 0

Count-free Weisfeiler–Leman and group isomorphism 无计数 Weisfeiler-Leman 和群同构

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-03 DOI: 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:<ul><li>Direct products of non-Abelian simple groups.</li><li>Coprime extensions, where the normal Hall subgroup is Abelian and the complement is an <math altimg="eq-00001.gif" display="inline" overflow="scroll"><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math>-generated solvable group with solvability class poly log log <math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>n</mi></math>. This notably includes instances where the complement is an <math altimg="eq-00003.gif" display="inline" overflow="scroll"><mi>O</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></math>-generated nilpotent group. This problem was previously known to be in <math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi mathvariant="sans-serif">P</mi></math> [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 <math altimg="eq-00005.gif" display="inline" overflow="scroll"><mi mathvariant="sans-serif">L</mi></math> [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].</li><li>Graphical groups of class <math altimg="eq-00006.gif" display="inline" overflow="scroll"><mn>2</mn></math> and exponent <math altimg="eq-00007.gif" display="inline" overflow="scroll"><mi>p</mi><mo>></mo><mn>2</mn></math> [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。

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引用次数: 0

Gapsets and the k-generalized Fibonacci sequences 缺口集和 k 个广义斐波那契序列

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-03 DOI: 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.

我们将数字半群的 Kunz 坐标术语引入间隙集，并将这一概念推广到 m-扩展。这样，我们就能识别间隙集，一般来说，也能识别 m-扩展与棋盘的倾斜；因此，我们介绍了这种识别的一些应用。此外，当多重性为 3 或 4 时，我们给出了具有固定属和深度的隙集数的明确公式，并在某些情况下给出了具有固定属和深度的隙集数的明确公式。

引用次数: 0

Bounding embedded singularities of Hilbert schemes of points on affine three space 仿射三空间上点的希尔伯特方案的边界嵌入奇点

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-03 DOI: 10.1142/s0218196724500140

Jen-Chieh Hsiao

The Hilbert scheme <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> of <math altimg="eq-00002.gif" display="inline" overflow="scroll"><mi>n</mi></math> points on <math altimg="eq-00003.gif" display="inline" overflow="scroll"><msup><mrow><mi>ℂ</mi></mrow><mrow><mn>3</mn></mrow></msup></math> can be expressed as the critical locus of a regular function on a smooth variety <math altimg="eq-00004.gif" display="inline" overflow="scroll"><mi mathvariant="cal">𝒳</mi></math>. Recent development in birational geometry suggests a study of singularities of the pair <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> using jet schemes. In this paper, we use a comparison between <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> and the scheme <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> of three commuting <math altimg="eq-00008.gif" display="inline" overflow="scroll"><mi>n</mi><mo stretchy="false">×</mo><mi>n</mi></math> matrices to estimate the log canonical threshold of <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>. As a consequence, we see that although both <math altimg="eq-00010.gif" display="inline" overflow="scroll"><mo>dim</mo><mi mathvariant="cal">𝒳</mi></math> and <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> have asymptotic growth <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>,

ℂ3上n个点的希尔伯特方案Hilbnℂ3可以表示为光滑品种𝒳上正则函数的临界点。双元几何的最新发展表明，可以利用喷流方案研究一对 (𝒳,Hilbnℂ3) 的奇点。在本文中，我们通过比较 Hilbnℂ3 与由三个换向 n×n 矩阵组成的方案 C3,n 来估计 (𝒳,Hilbnℂ3) 的对数规范阈值。结果我们发现，虽然 dim𝒳 和 dimHilbnℂ3 的渐近增长都是 O(n2)，但 Hilbnℂ3 上任意点的最大多重性最多只有线性增长 O(n)。

引用次数: 0

On automorphisms of certain free nilpotent-by-abelian Lie algebras 论某些自由零能旁阿贝尔李代数的自形性

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-04-03 DOI: 10.1142/s0218196724500097

C. E. Kofinas, A. I. Papistas

For a positive integer $n \geq 4$ , let $R_{n}$ 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 $Aut (R_{n})$ generated by the tame automorphisms and a countably infinite set of explicitly given automorphisms of $R_{n}$ is dense in $Aut (R_{n})$ with respect to the formal power series topology.

对于正整数 n≥4，设 Rn 是秩为 n 的自由（2 类无势）旁阿贝尔和旁无边际（2 类无势）的李代数。我们证明，就形式幂级数拓扑而言，由 Rn 的驯服自形和一组可数无限的明给自形生成的 Aut(Rn) 子群在 Aut(Rn) 中是密集的。

引用次数: 0

Products of traceless and semi-traceless matrices over division rings and their applications 分割环上无踪和半无踪矩阵的乘积及其应用

IF 0.8 2区数学 Q3 MATHEMATICS

International Journal of Algebra and Computation

Pub Date : 2024-03-26 DOI: 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.

在本文中，我们证明了划分环上的每个矩阵都可以表示为最多 10 个无踪矩阵的乘积，以及最多 4 个半无踪矩阵的乘积。通过应用这一结果和迄今为止获得的其他结果，我们证明了某些代数的元素具有一些相当有趣的非难分解，即分解为非交换多项式的图像的乘积。

引用次数: 0

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