Pub Date : 2022-06-17DOI: 10.1142/s0218196723500571
Eduardo Silva
For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $Awr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.
{"title":"Dead ends on wreath products and lamplighter groups","authors":"Eduardo Silva","doi":"10.1142/s0218196723500571","DOIUrl":"https://doi.org/10.1142/s0218196723500571","url":null,"abstract":"For any finite group $A$ and any finitely generated group $B$, we prove that the corresponding lamplighter group $Awr B$ admits a standard generating set with unbounded depth, and that if $B$ is abelian then the above is true for every standard generating set. This generalizes the case where $B=mathbb{Z}$ together with its cyclic generator due to Cleary and Taback. When $B=H*K$ is the free product of two finite groups $H$ and $K$, we characterize which standard generators of the associated lamplighter group have unbounded depth in terms of a geometrical constant related to the Cayley graphs of $H$ and $K$. In particular, we find differences with the one-dimensional case: the lamplighter group over the free product of two sufficiently large finite cyclic groups has uniformly bounded depth with respect to some standard generating set.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44320880","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 : 2022-05-24DOI: 10.1142/s0218196723500479
Daniel Chen, Xiaojin Zhang
Let $Lambda$ be a radical square zero algebra of a Dynkin quiver and let $Gamma$ be the Auslander algebra of $Lambda$. Then the number of tilting right $Gamma$-modules is $2^{m-1}$ if $Lambda$ is of $A_{m}$ type for $mgeq 1$. Otherwise, the number of tilting right $Gamma$-modules is $2^{m-3}times14$ if $Lambda$ is either of $D_{m}$ type for $mgeq 4$ or of $E_{m}$ type for $m=6,7,8$.
{"title":"On the Number of Tilting Modules Over a Class Of Auslander Algebras","authors":"Daniel Chen, Xiaojin Zhang","doi":"10.1142/s0218196723500479","DOIUrl":"https://doi.org/10.1142/s0218196723500479","url":null,"abstract":"Let $Lambda$ be a radical square zero algebra of a Dynkin quiver and let $Gamma$ be the Auslander algebra of $Lambda$. Then the number of tilting right $Gamma$-modules is $2^{m-1}$ if $Lambda$ is of $A_{m}$ type for $mgeq 1$. Otherwise, the number of tilting right $Gamma$-modules is $2^{m-3}times14$ if $Lambda$ is either of $D_{m}$ type for $mgeq 4$ or of $E_{m}$ type for $m=6,7,8$.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45072948","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}
A general implementation is presented for constructing the relation between the conservation laws for partial differential equations and the Lie algebra. This construction does not require the use of existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit formula is derived which yields a conservation law for each solution of the determining system. A simulation of this combination to solve partial differential equation is elaborated by an application on Burger’s equation which shows several results. General behavior of the distribution function for conservation laws of these equations are obtained and shown.
{"title":"Lie algebras and hyperbolic conservation laws","authors":"Yacine Benhadid, Yousuf Alkhezi","doi":"10.12988/ija.2022.91731","DOIUrl":"https://doi.org/10.12988/ija.2022.91731","url":null,"abstract":"A general implementation is presented for constructing the relation between the conservation laws for partial differential equations and the Lie algebra. This construction does not require the use of existence of a variational principle and reduces the calculation of conservation laws to solving a system of linear determining equations similar to that for finding symmetries. An explicit formula is derived which yields a conservation law for each solution of the determining system. A simulation of this combination to solve partial differential equation is elaborated by an application on Burger’s equation which shows several results. General behavior of the distribution function for conservation laws of these equations are obtained and shown.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"28 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85435858","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}
The aim of this paper is to focus on the GroneMerris Theorem and Brouwers Conjecture and give the results of some famous graphs. For example, a threshold graph, a star graph, ( K 4 − e ) + K 2 . Moreover, we investigated some common properties between nonplanar graphs K 3 , 3 , K 5 in this regard. We have relied on [1], [6] and [9].
本文主要讨论了GroneMerris定理和browwers猜想，并给出了一些著名图的结果。例如，一个阈值图，一个星图，(K 4−e) + K 2。此外，我们还研究了非平面图k3、k3、k5之间的一些共同性质。我们依赖于[1]，[6]和[9]。
{"title":"Laplacian spectrum and Brouwer's conjecture of a graph","authors":"Nada Alnufaei","doi":"10.12988/ija.2022.91724","DOIUrl":"https://doi.org/10.12988/ija.2022.91724","url":null,"abstract":"The aim of this paper is to focus on the GroneMerris Theorem and Brouwers Conjecture and give the results of some famous graphs. For example, a threshold graph, a star graph, ( K 4 − e ) + K 2 . Moreover, we investigated some common properties between nonplanar graphs K 3 , 3 , K 5 in this regard. We have relied on [1], [6] and [9].","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"19 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76863452","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":"Ranks in elliptic curves of the forms y^2=x^3+Ax^2+Bx","authors":"Shin-wook Kim","doi":"10.12988/ija.2022.91726","DOIUrl":"https://doi.org/10.12988/ija.2022.91726","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"24 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77419897","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}
In this paper, we classify R-train algebras of rank 3 and dimension 6 of type (4, 2) for which dim U2 = dim UV = 1 and U3 = 0. For that, we use some results that we had proved in [3] and in [4]. Mathematics Subject Classification: 17A30, 17C27
本文对(4,2)型3阶6维r -训练代数进行了分类，其中dim U2 = dim UV = 1, U3 = 0。为此，我们使用了在[3]和[4]中已经证明的一些结果。数学学科分类:17A30, 17C27
{"title":"R-train algebras of rank 3, type (4, 2) with dim U2 = dim UV = 1 and U3 = 0","authors":"R. Arbach, Luis Antonio Fernandes de Oliveira","doi":"10.12988/ija.2022.91717","DOIUrl":"https://doi.org/10.12988/ija.2022.91717","url":null,"abstract":"In this paper, we classify R-train algebras of rank 3 and dimension 6 of type (4, 2) for which dim U2 = dim UV = 1 and U3 = 0. For that, we use some results that we had proved in [3] and in [4]. Mathematics Subject Classification: 17A30, 17C27","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"97 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81287226","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}
The present work concerns certain aspects of Lie algebras, Lie triple systems and Jordan triple systems. We summarize the latest results on the universal enveloping algebras of Lie algebras and Lie triple systems. In particular, we study a Casimir element as the element in the center of the universal enveloping algebras. Using this element, we characterize the semi-simple Lie triple systems among the quadratic Lie triple systems. We define Jordan’s triple systems relationship with algebras and Lie algebras. Finally, we prove some theorems, examples and facts on all of the above. Mathematics Subject Classification: 17C50, 17B60, 17B20, 17B05
{"title":"On representation theory of Lie triple systems","authors":"A. Alshahrani","doi":"10.12988/ija.2022.91720","DOIUrl":"https://doi.org/10.12988/ija.2022.91720","url":null,"abstract":"The present work concerns certain aspects of Lie algebras, Lie triple systems and Jordan triple systems. We summarize the latest results on the universal enveloping algebras of Lie algebras and Lie triple systems. In particular, we study a Casimir element as the element in the center of the universal enveloping algebras. Using this element, we characterize the semi-simple Lie triple systems among the quadratic Lie triple systems. We define Jordan’s triple systems relationship with algebras and Lie algebras. Finally, we prove some theorems, examples and facts on all of the above. Mathematics Subject Classification: 17C50, 17B60, 17B20, 17B05","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"73 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75692384","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}
In this paper, we study the reduced rings arising from Morita context M ( A, B ) = ( A, M, N, B, ϕ, ψ ). Necessary and sufficient conditions are investigated for the Morita ring R to be reduced. In particular, the reduced modules over Morita context rings are characterized.
在本文中，我们研究了由Morita上下文M (A, B) = (A, M, N, B， φ， ψ)产生的约简环。研究了减小森田环R的充分必要条件。特别地，对Morita上下文环上的约简模进行了表征。
{"title":"Reduced rings and modules arising from Morita contexts","authors":"Qingbing Xu, Yang Liu, M. Munir, Kausar Nasreen","doi":"10.12988/ija.2022.91725","DOIUrl":"https://doi.org/10.12988/ija.2022.91725","url":null,"abstract":"In this paper, we study the reduced rings arising from Morita context M ( A, B ) = ( A, M, N, B, ϕ, ψ ). Necessary and sufficient conditions are investigated for the Morita ring R to be reduced. In particular, the reduced modules over Morita context rings are characterized.","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"2013 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87908109","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}
E. Salinas-Hernández, G. Ares de Parga, J. A. Martinez-Nuno
{"title":"Direct calculation for determinant main blocks of 2x2","authors":"E. Salinas-Hernández, G. Ares de Parga, J. A. Martinez-Nuno","doi":"10.12988/ija.2022.91715","DOIUrl":"https://doi.org/10.12988/ija.2022.91715","url":null,"abstract":"","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"27 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82636212","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}
In 1956, P.M. Cohn gave necessary and sufficient conditions for a semigroup to be embeddable in a left simple semigroup. The main purpose of this paper is to show how to construct a semigroup which is embeddable in an idempotent-free left simple semigroup. Mathematics Subject Classification: 20M10
{"title":"On an embedding theorem of Cohn","authors":"A. Nagy","doi":"10.12988/ija.2022.91647","DOIUrl":"https://doi.org/10.12988/ija.2022.91647","url":null,"abstract":"In 1956, P.M. Cohn gave necessary and sufficient conditions for a semigroup to be embeddable in a left simple semigroup. The main purpose of this paper is to show how to construct a semigroup which is embeddable in an idempotent-free left simple semigroup. Mathematics Subject Classification: 20M10","PeriodicalId":13756,"journal":{"name":"International Journal of Algebra and Computation","volume":"18 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72786287","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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