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":"2016 1","pages":""},"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
<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
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":"146 1","pages":""},"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":"29 1","pages":""},"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 : 2023-12-08DOI: 10.1142/s0218196723500698
M. Al-Tahan, B. Davvaz, P. Harikrishnan, P. Pallavi
{"title":"Subpolygroup commutativity degree of finite extension polygroup","authors":"M. Al-Tahan, B. Davvaz, P. Harikrishnan, P. Pallavi","doi":"10.1142/s0218196723500698","DOIUrl":"https://doi.org/10.1142/s0218196723500698","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"253 7","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139011475","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 : 2023-12-08DOI: 10.1142/s0218196723500686
H. A. Çoban
{"title":"Detecting similarities of rational plane curves using complex differential invariants","authors":"H. A. Çoban","doi":"10.1142/s0218196723500686","DOIUrl":"https://doi.org/10.1142/s0218196723500686","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"130 ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139011530","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 : 2023-12-01DOI: 10.1142/s0218196723990011
{"title":"Author index Volume 33 (2023)","authors":"","doi":"10.1142/s0218196723990011","DOIUrl":"https://doi.org/10.1142/s0218196723990011","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"21 4","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139194284","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 : 2023-11-10DOI: 10.1142/s0218196723500674
Jesse Lansdown
Using the classification of transitive groups of degree $n$, for $2 leqslant n leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 leqslant n leqslant 48$. The results of this classification are made available as a supplementary database.
{"title":"A census of small Schurian association schemes","authors":"Jesse Lansdown","doi":"10.1142/s0218196723500674","DOIUrl":"https://doi.org/10.1142/s0218196723500674","url":null,"abstract":"Using the classification of transitive groups of degree $n$, for $2 leqslant n leqslant 48$, we classify the Schurian association schemes of order $n$, and as a consequence, the transitive groups of degree $n$ that are $2$-closed. In addition, we compute the character table of each association scheme and provide a census of important properties. Finally, we compute the $2$-closure of each transitive group of degree $n$, for $2 leqslant n leqslant 48$. The results of this classification are made available as a supplementary database.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":" 11","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191733","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 : 2023-11-10DOI: 10.1142/s0218196723500662
C. E. Kofinas
{"title":"Symmetric polynomials in free centre-by-metabelian lie algebras of rank 2","authors":"C. E. Kofinas","doi":"10.1142/s0218196723500662","DOIUrl":"https://doi.org/10.1142/s0218196723500662","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":" 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191022","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 : 2023-11-09DOI: 10.1142/s0218196723500650
Claudemir Fideles, Alan Guimaraes
Let [Formula: see text] be the infinite-dimensional Grassmann algebra over an infinite field [Formula: see text] of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of [Formula: see text] equipped with its [Formula: see text] and [Formula: see text]-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.
设[公式:见文]为特征不等于2的无限域上的无限维Grassmann代数[公式:见文]。本文的主要目的是描述[公式:见文]的分级多项式恒等式的[理想]和[公式:见文]的中心多项式的[公式:见文]空间及其[公式:见文]和[公式:见文]诱导的[公式:见文]分级。因此，我们推广了[A]的结果。brand， C. Fidelis和A. guimar，[公式:见文本]-格拉斯曼代数的完全支持度评分，数学学报，31 (2022):332-353;C. Fidelis, A. guimar es和P. Koshlukov，关于[公式:见文本]的注解-关于Grassmann代数和初等数论的评分，线性多线性代数，71(7)(2023)1244-1264]。
{"title":"ℤ-Gradings on the Grassmann Algebra Over Infinite Fields: Graded Identities and Central Polynomials","authors":"Claudemir Fideles, Alan Guimaraes","doi":"10.1142/s0218196723500650","DOIUrl":"https://doi.org/10.1142/s0218196723500650","url":null,"abstract":"Let [Formula: see text] be the infinite-dimensional Grassmann algebra over an infinite field [Formula: see text] of characteristic different from 2. The main purpose of this paper is to describe the [Formula: see text]-ideal of the graded polynomial identities and the [Formula: see text]-space of the central polynomials of [Formula: see text] equipped with its [Formula: see text] and [Formula: see text]-induced [Formula: see text]-gradings. Therefore, we generalize the results of [A. Brandão, C. Fidelis and A. Guimarães, [Formula: see text]-gradings of full support on the Grassmann algebra, J. Algebra 601 (2022) 332–353; C. Fidelis, A. Guimarães and P. Koshlukov, A note on [Formula: see text]-gradings on the Grassmann algebra and elementary number theory, Linear Multilinear Algebra 71(7) (2023) 1244–1264] in the PI theory context.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":" 85","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191123","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 : 2023-09-22DOI: 10.1142/s0218196723500649
Alexander G. Melnikov, Keng Meng Ng
. We compare several natural notions of effective presentability of a topological space up to homeomorphism. We note that every left-c.e. (lower-semicomputable) Stone space is homeomorphic to a computable one. In contrast, we produce an example of a locally compact, left-c.e. space that is not homeomorphic to any computable Polish space. We apply a similar technique to produce examples of computable topological spaces not homeomorphic to any right-c.e. (upper-semicomputable) Polish space, and indeed to any arith-metical or even analytical Polish space. We then apply our techniques to totally disconnected locally compact (tdlc) groups. We prove that every effectively locally compact tdlc group is topologically isomorphic to a computable tdlc group; all notions will be clarified. The result is perhaps unexpected since the hypothesis of the theorem may seem rather weak.
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