The aim of this paper is to consider Leibniz algebras, whose principal ideals are finite dimensional. We prove that the derived ideal of L has finite dimension if every principal ideal of a Leibniz algebra L has dimension at most b, where b is a fixed positive integer.
{"title":"Ideally finite Leibniz algebras","authors":"L. A. Kurdachenko, I. Ya. Subbotin","doi":"10.12958/adm2139","DOIUrl":"https://doi.org/10.12958/adm2139","url":null,"abstract":"The aim of this paper is to consider Leibniz algebras, whose principal ideals are finite dimensional. We prove that the derived ideal of L has finite dimension if every principal ideal of a Leibniz algebra L has dimension at most b, where b is a fixed positive integer.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136305700","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of a book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap "Bieberbach groups and flat manifolds" was published.
{"title":"Geometry of Crystallographic Groups","authors":"A. Szczepański","doi":"10.1142/8519","DOIUrl":"https://doi.org/10.1142/8519","url":null,"abstract":"Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. They got their name, because in three dimensions they occur as the symmetry groups of a crystal (which we imagine to extend to infinity in all directions). The book is divided into two parts. In the first part, the basic theory of crystallographic groups is developed from the very beginning, while in the second part, more advanced and more recent topics are discussed. So the first part of the book should be usable as a textbook, while the second part is more interesting to researchers in the field. There are short introductions to the theme before every chapter. At the end of a book is a list of conjectures and open problems. Moreover there are three appendices. The last one gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group. This volume omits topics about generalization of crystallographic groups to nilpotent or solvable world and classical crystallography. We want to emphasize that most theorems and facts presented in the second part are from the last two decades. This is after the book of L Charlap \"Bieberbach groups and flat manifolds\" was published.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2012-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121564026","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Definitions, Theorems, and Proofs Arithmetic Graphs Circuits Trees Basics of Sets Relations and Databases Induction Reachability Games on Graphs Functions and Transitions Propositional Logic Finite Automata Regular Expressions Counting Probability.
{"title":"Lectures on Discrete Mathematics for Computer Science","authors":"B. Khoussainov, Nodira Khoussainova","doi":"10.1142/8084","DOIUrl":"https://doi.org/10.1142/8084","url":null,"abstract":"Definitions, Theorems, and Proofs Arithmetic Graphs Circuits Trees Basics of Sets Relations and Databases Induction Reachability Games on Graphs Functions and Transitions Propositional Logic Finite Automata Regular Expressions Counting Probability.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2012-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114360304","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Linear Algebra and Projective Geometry Buildings and Grassmannians Classical Grassmannians Polar and Half-Spin Grassmannians.
线性代数和射影几何建筑和格拉斯曼人经典格拉斯曼人极地和半旋格拉斯曼人。
{"title":"Grassmannians of Classical Buildings","authors":"M. Pankov","doi":"10.1142/7844","DOIUrl":"https://doi.org/10.1142/7844","url":null,"abstract":"Linear Algebra and Projective Geometry Buildings and Grassmannians Classical Grassmannians Polar and Half-Spin Grassmannians.","PeriodicalId":364397,"journal":{"name":"Algebra and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2008-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116109361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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