Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ for all $a,b,cin L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,bin L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of non-nilpotent three-dimensional Leibniz algebras.
{"title":"Automorphism groups of some non-nilpotent Leibniz algebras","authors":"L. A. Kurdachenko, P. Minaiev, O. Pypka","doi":"10.15421/242409","DOIUrl":"https://doi.org/10.15421/242409","url":null,"abstract":"Let $L$ be an algebra over a field $F$ with the binary operations $+$ and $[,]$. Then $L$ is called a left Leibniz algebra if it satisfies the left Leibniz identity: $[a,[b,c]]=[[a,b],c]+[b,[a,c]]$ for all $a,b,cin L$. A linear transformation $f$ of $L$ is called an endomorphism of $L$, if $f([a,b])=[f(a),f(b)]$ for all elements $a,bin L$. A bijective endomorphism of $L$ is called an automorphism of $L$. It is easy to show that the set of all automorphisms of Leibniz algebra is a group with respect to the operation of multiplication of automorphisms. The description of the structure of the automorphism groups of Leibniz algebras is one of the natural and important problems of the general Leibniz algebra theory. The main goal of this article is to describe the structure of the automorphism group of a certain type of non-nilpotent three-dimensional Leibniz algebras.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"104 38","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667399","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}
One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.
{"title":"On Poisson (2-3)-algebras which are finite-dimensional over the center","authors":"P. Minaiev, O. Pypka, I. Shyshenko","doi":"10.15421/242411","DOIUrl":"https://doi.org/10.15421/242411","url":null,"abstract":"One of the classic results of group theory is the so-called Schur theorem. It states that if the central factor-group $G/zeta(G)$ of a group $G$ is finite, then its derived subgroup $[G,G]$ is also finite. This result has numerous generalizations and modifications in group theory. At the same time, similar investigations were conducted in other algebraic structures, namely in modules, linear groups, topological groups, $n$-groups, associative algebras, Lie algebras, Lie $n$-algebras, Lie rings, Leibniz algebras. In 2021, L.A. Kurdachenko, O.O. Pypka and I.Ya. Subbotin proved an analogue of Schur theorem for Poisson algebras: if the center of the Poisson algebra $P$ has finite codimension, then $P$ includes an ideal $K$ of finite dimension such that $P/K$ is abelian. In this paper, we continue similar studies for another algebraic structure. An analogue of Schur theorem for Poisson (2-3)-algebras is proved.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":" July","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141669803","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}
Matroid is defined as a pair $(X,mathcal{I})$, where $X$ is a nonempty finite set, and $mathcal{I}$ is a nonempty set of subsets of $X$ that satisfies the Hereditary Axiom and the Augmentation Axiom. The paper investigates for which semigroups (primarily finite) $S$, the pair $(widehat{S}, mathcal{I})$ will be a matroid.
{"title":"Matroids related to groups and semigroups","authors":"D.I. Bezushchak","doi":"10.15421/242309","DOIUrl":"https://doi.org/10.15421/242309","url":null,"abstract":"Matroid is defined as a pair $(X,mathcal{I})$, where $X$ is a nonempty finite set, and $mathcal{I}$ is a nonempty set of subsets of $X$ that satisfies the Hereditary Axiom and the Augmentation Axiom. The paper investigates for which semigroups (primarily finite) $S$, the pair $(widehat{S}, mathcal{I})$ will be a matroid.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139157054","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}
Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.
{"title":"On invariant ideals in group rings of torsion-free minimax nilpotent groups","authors":"A. Tushev","doi":"10.15421/242315","DOIUrl":"https://doi.org/10.15421/242315","url":null,"abstract":"Let $k$ be a field and let $N$ be a nilpotent minimax torsion-free group acted by a solvable group of operators $G$ of finite rank. In the presented paper we study properties of some types of $G$-invariant ideals of the group ring $kN$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"15 12","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139156799","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}
There have been various studies on star-Lindelöfness but they always explain it in terms of open coverings. So, we have demonstrated in this study a connection between star-Lindelöfness and the family of closed sets that resembles countable intersection property of Lindelöf space. We show that a topological space $X$ is star-Lindelöf if and only if every closed subset's family of $X$ not having the modified non-countable intersection property have non-empty intersection.
{"title":"A Countable Intersection Like Characterization of Star-Lindelöf Spaces","authors":"P. Bal","doi":"10.15421/242308","DOIUrl":"https://doi.org/10.15421/242308","url":null,"abstract":"There have been various studies on star-Lindelöfness but they always explain it in terms of open coverings. So, we have demonstrated in this study a connection between star-Lindelöfness and the family of closed sets that resembles countable intersection property of Lindelöf space. We show that a topological space $X$ is star-Lindelöf if and only if every closed subset's family of $X$ not having the modified non-countable intersection property have non-empty intersection.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"13 23","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139156857","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}
An attempt is made to introduce and use operational techniques to study about a new sequence of functions containing generalized Jacobi polynomial. Some generating relations, finite summation formulae, explicit representation of a sequence of function $S_{n,tau ,k}^{(alpha ,beta ,gamma ,delta )} (x;a,u,v)$ associated with the generalized Jacobi polynomial $P_{n,,tau }^{left( {alpha ,,gamma ,,beta } right)} (x)$ have been deduced.
{"title":"A Note on Sequence of Functions associated with the Generalized Jacobi polynomial","authors":"D. Waghela, S.B. Rao","doi":"10.15421/242316","DOIUrl":"https://doi.org/10.15421/242316","url":null,"abstract":"An attempt is made to introduce and use operational techniques to study about a new sequence of functions containing generalized Jacobi polynomial. Some generating relations, finite summation formulae, explicit representation of a sequence of function $S_{n,tau ,k}^{(alpha ,beta ,gamma ,delta )} (x;a,u,v)$ associated with the generalized Jacobi polynomial $P_{n,,tau }^{left( {alpha ,,gamma ,,beta } right)} (x)$ have been deduced.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"88 2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139155933","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}
For every odd prime $p$ we construct two $p$-automata with 14 inner states and prove that the group generated by 2 automaton permutations defined at their states is a free group of rank 2.
{"title":"Free groups defined by finite $p$-automata","authors":"A. Krenevych, A. Oliynyk","doi":"10.15421/242314","DOIUrl":"https://doi.org/10.15421/242314","url":null,"abstract":"For every odd prime $p$ we construct two $p$-automata with 14 inner states and prove that the group generated by 2 automaton permutations defined at their states is a free group of rank 2.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"12 13","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139156698","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}
The paper considers the problem of establishing the convergence criteria of the branched continued fraction expansion of the ratio of Horn's hypergeometric functions $H_4$. To solve it, the technique of expanding the domain of convergence of the branched continued fraction from the known small domain of convergence to a wider domain of convergence is used. For the real and complex parameters of the Horn hypergeometric function $H_4$, a number of convergence criteria of the branched continued fraction expansion under certain conditions to its coefficients in various unbounded domains of the space have been established.
{"title":"On the domains of convergence of the branched continued fraction expansion of ratio $H_4(a,d+1;c,d;mathbf{z})/H_4(a,d+2;c,d+1;mathbf{z})$","authors":"R. Dmytryshyn, I.-A.V. Lutsiv, O.S. Bodnar","doi":"10.15421/242311","DOIUrl":"https://doi.org/10.15421/242311","url":null,"abstract":"The paper considers the problem of establishing the convergence criteria of the branched continued fraction expansion of the ratio of Horn's hypergeometric functions $H_4$. To solve it, the technique of expanding the domain of convergence of the branched continued fraction from the known small domain of convergence to a wider domain of convergence is used. For the real and complex parameters of the Horn hypergeometric function $H_4$, a number of convergence criteria of the branched continued fraction expansion under certain conditions to its coefficients in various unbounded domains of the space have been established.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"68 3","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139155077","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}
For given unit vectors $x_1, cdots, x_n$ of a real Banach space $E,$ we define $$NA({mathcal L}(^nE))(x_1, cdots, x_n)={Tin {mathcal L}(^nE): |T(x_1, cdots, x_n)|=|T|=1},$$ where ${mathcal L}(^nE)$ denotes the Banach space of all continuous $n$-linear forms on $E$ endowed with the norm $|T|=sup_{|x_k|=1, 1leq kleq n}{|T(x_1, ldots, x_n)|}$.In this paper, we classify $NA({mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$ for unit vectors $Z_1, Z_2in d_{*}(1, w)^2,$ where $d_{*}(1, w)^2=mathbb{R}^2$ with the norm of weight $0
{"title":"Norm attaining bilinear forms of ${mathcal L}(^2 d_{*}(1, w)^2)$ at given vectors","authors":"S.G. Kim","doi":"10.15421/242313","DOIUrl":"https://doi.org/10.15421/242313","url":null,"abstract":"For given unit vectors $x_1, cdots, x_n$ of a real Banach space $E,$ we define $$NA({mathcal L}(^nE))(x_1, cdots, x_n)={Tin {mathcal L}(^nE): |T(x_1, cdots, x_n)|=|T|=1},$$ where ${mathcal L}(^nE)$ denotes the Banach space of all continuous $n$-linear forms on $E$ endowed with the norm $|T|=sup_{|x_k|=1, 1leq kleq n}{|T(x_1, ldots, x_n)|}$.In this paper, we classify $NA({mathcal L}(^2 d_{*}(1, w)^2))(Z_1, Z_2)$ for unit vectors $Z_1, Z_2in d_{*}(1, w)^2,$ where $d_{*}(1, w)^2=mathbb{R}^2$ with the norm of weight $0<w<1$ endowed with $|(x, y)|_{d_*(1, w)}=maxBig{|x|, |y|, frac{|x|+|y|}{1+w}Big}$.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"8 4","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139156780","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}
We describe derivations of several important associative and Lie rings of infinite matrices over general rings of coefficients.
我们描述了一般系数环上无限矩阵的几个重要关联环和李环的推导。
{"title":"Derivations of rings of infinite matrices","authors":"O. Bezushchak","doi":"10.15421/242310","DOIUrl":"https://doi.org/10.15421/242310","url":null,"abstract":"We describe derivations of several important associative and Lie rings of infinite matrices over general rings of coefficients.","PeriodicalId":52827,"journal":{"name":"Researches in Mathematics","volume":"23 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139318123","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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