In this paper an extensive study of the concepts of generalized Green’s relations and GV -semigroups without order to ordered semigroups have been given. Our approach allows one to see the nature of generalized Green’s relations in the class of GV -ordered semigroups. Moreover we show that an ordered semigroup S is a GV -ordered semigroup if and only if S is a complete semilattice of completely π-regular and Archimedean ordered semigroups.
{"title":"Generalized Green's relations and GV-ordered semigroups","authors":"Shauli Sadhya, K. Hansda","doi":"10.56415/qrs.v30.14","DOIUrl":"https://doi.org/10.56415/qrs.v30.14","url":null,"abstract":"In this paper an extensive study of the concepts of generalized Green’s relations and GV -semigroups without order to ordered semigroups have been given. Our approach allows one to see the nature of generalized Green’s relations in the class of GV -ordered semigroups. Moreover we show that an ordered semigroup S is a GV -ordered semigroup if and only if S is a complete semilattice of completely π-regular and Archimedean ordered semigroups.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46979088","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}
Pre-anti-flexible family algebras are introduced and used to define and describe the notions of Ωc-relative anti-flexible algebras, left and right pre-Lie family algebras and Ωc-relative Lie algebras. The notion of Ωc-relative pre-anti-flexible algebras are introduced and also used to characterize pre-anti-flexible family algebras, left and right pre-Lie family algebras and significant identities associated to these algebraic structures are provided. Finally, a generalization of the Rota-Baxter operators defined on an Ωc-relative anti-flexible algebra is introduced and it is also proved that both Rota-Baxter operators and its generalization provide Ωc-relative pre-antiflexible algebras structures and related consequences are derived.
{"title":"Relative (pre-)anti-flexible algebrasnand associated algebraic structures","authors":"Mafoya Landry Dassoundo","doi":"10.56415/qrs.v30.03","DOIUrl":"https://doi.org/10.56415/qrs.v30.03","url":null,"abstract":"Pre-anti-flexible family algebras are introduced and used to define and describe the notions of Ωc-relative anti-flexible algebras, left and right pre-Lie family algebras and Ωc-relative Lie algebras. The notion of Ωc-relative pre-anti-flexible algebras are introduced and also used to characterize pre-anti-flexible family algebras, left and right pre-Lie family algebras and significant identities associated to these algebraic structures are provided. Finally, a generalization of the Rota-Baxter operators defined on an Ωc-relative anti-flexible algebra is introduced and it is also proved that both Rota-Baxter operators and its generalization provide Ωc-relative pre-antiflexible algebras structures and related consequences are derived.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46840570","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}
A simplicial group is a simplicial object in the category of groups. A very nice application of simplicial group which is simplicial polygroup is given in this paper. Using polygroups instead of groups, we already had very good results from the well known properties due to Loday. Loday proved that a crossed module, a cat1-group, a group object in the category of categories and a simplicial group whose Moore complex is of length one are equivalent. Using Loday’s idea we present a functor from the category of groups to the category of polygroups and the simplicial groups to the simplicial polygroups. We show that there exist a functor from the category of cat1-polygroups to the category of groups and the category of groups to the category of polygroups. We also prove that the category of simplicial groups is equivalent to the category of simplicial polygroups and the category of simplicial polygroups with generalized Moore complex with of length one is equivalent to the category of polygroups.
{"title":"Simplicial polygroups and the generalized Moore complexes","authors":"Davvaz Bijan, Alp Murat","doi":"10.56415/qrs.v30.04","DOIUrl":"https://doi.org/10.56415/qrs.v30.04","url":null,"abstract":"A simplicial group is a simplicial object in the category of groups. A very nice application of simplicial group which is simplicial polygroup is given in this paper. Using polygroups instead of groups, we already had very good results from the well known properties due to Loday. Loday proved that a crossed module, a cat1-group, a group object in the category of categories and a simplicial group whose Moore complex is of length one are equivalent. Using Loday’s idea we present a functor from the category of groups to the category of polygroups and the simplicial groups to the simplicial polygroups. We show that there exist a functor from the category of cat1-polygroups to the category of groups and the category of groups to the category of polygroups. We also prove that the category of simplicial groups is equivalent to the category of simplicial polygroups and the category of simplicial polygroups with generalized Moore complex with of length one is equivalent to the category of polygroups.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42608027","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}
Four-dimensional finite non-commutative associative algebras represent practical interest as algebraic support of post-quantum digital signature algorithms, especially algebras with two sided global unit, set by sparse basis vectors multiplication tables. A new algebra of the latter type, set over the field GF(p), is proposed and its structure is investigated. The studied algebra is described as a set of p2 + p + 1 commutative subalgebras of three different types. All subalgebras intersect strictly in the subset of scalar vectors. Formulas are derived for the number of subalgebras of each type.
四维有限非交换关联代数作为后量子数字签名算法的代数支持具有实际意义，特别是具有两侧全局单位的代数，由稀疏基向量乘法表集合。提出了一种新的后一种类型的代数，集在域GF(p)上，并研究了它的结构。所研究的代数被描述为三种不同类型的p2 + p + 1交换子代数的集合。所有子代数在标量向量的子集中严格相交。导出了每种类型的子代数的数目的公式。
{"title":"Structure of a finite non-commutative algebra set by a sparse multiplication table","authors":"D. Moldovyan, A. Moldovyan, N. Moldovyan","doi":"10.56415/qrs.v30.11","DOIUrl":"https://doi.org/10.56415/qrs.v30.11","url":null,"abstract":"Four-dimensional finite non-commutative associative algebras represent practical interest as algebraic support of post-quantum digital signature algorithms, especially algebras with two sided global unit, set by sparse basis vectors multiplication tables. A new algebra of the latter type, set over the field GF(p), is proposed and its structure is investigated. The studied algebra is described as a set of p2 + p + 1 commutative subalgebras of three different types. All subalgebras intersect strictly in the subset of scalar vectors. Formulas are derived for the number of subalgebras of each type.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45331117","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}
In this paper, the concepts of irreducible and strongly irreducible pseudo symmetric ideals in a partially ordered ternary semigroup are introduced. We also studied some interesting properties of irreducible and strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup and prove that the space of strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup is topologized.
{"title":"On irreducible pseudo symmetric ideals of a partially ordered ternary semigroup","authors":"Dattatraya Shinde, Machchhindra Gophane","doi":"10.56415/qrs.v30.15","DOIUrl":"https://doi.org/10.56415/qrs.v30.15","url":null,"abstract":"In this paper, the concepts of irreducible and strongly irreducible pseudo symmetric ideals in a partially ordered ternary semigroup are introduced. We also studied some interesting properties of irreducible and strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup and prove that the space of strongly irreducible pseudo symmetric ideals of a partially ordered ternary semigroup is topologized.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41503223","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 construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gröbner-Shirshov bases for replicated algebras, we obtain a linear basis. As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNNextension.
{"title":"Operadic approach to HNN-extensions of Leibniz algebras","authors":"Georg Klein, Chia Zargeh","doi":"10.56415/qrs.v30.08","DOIUrl":"https://doi.org/10.56415/qrs.v30.08","url":null,"abstract":"We construct HNN-extensions of Lie di-algebras in the variety of di-algebras and provide a presentation for the replicated HNN-extension of a Lie di-algebras. Then, by applying the method of Gröbner-Shirshov bases for replicated algebras, we obtain a linear basis. As an application of HNN-extensions, we prove that Lie di-algebras are embedded in their HNNextension.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46820911","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 intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).
{"title":"On the intersection ideal graph of semigroups","authors":"Barkha Baloda, J. Kumar","doi":"10.56415/qrs.v31.01","DOIUrl":"https://doi.org/10.56415/qrs.v31.01","url":null,"abstract":"The intersection ideal graph Γ(S) of a semigroup S is a simple undirected graph whose vertices are all nontrivial left ideals of S and two distinct left ideals I, J are adjacent if and only if their intersection is nontrivial. In this paper, we investigate the connectedness of Γ(S). We show that if Γ(S) is connected, then the diameter of Γ(S) is at most two. Further, we classify the semigroups S in terms of their ideals such that the diameter of Γ(S) is two. We obtain the domination number, independence number, girth and the strong metric dimension of Γ(S). We have also investigated the completeness, planarity and perfectness of Γ(S). We show that if S is a completely simple semigroup, then Γ(S) is weakly perfect. More over, in this article, we give an upper bound of the chromatic number of Γ(S). Finally, if S is the union of n minimal left ideals, then we obtain the metric dimension and the automorphism group of Γ(S).","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47418318","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}
Mahdavi Soheila, Ashrafi Ali-Reza, Salahshour Mohammad A.
Suppose that (T;*) is a groupoid with a left identity such that each element a 2 T has a left inverse. Then T is called a gyrogroup if and only if (i) there exists a function gyr : T x T -Aut(T) such that for all a; b; c 2 T, a * (b * c) = (a * b) ? gyr[a; b]c, where gyr[a; b]c = gyr(a; b)(c); and (ii) for all a; b 2 T, gyr[a; b] = gyr[a ? b; b]. In this paper, the structure of normal subgyrogroups of certain gyrogroups are investigated.
假设(T;*)是一个具有左单位元的群，使得2t的每个元素都有一个左逆。那么当且仅当(i)存在一个函数gyr: T x T -Aut(T)使得对于所有a;b;c2t, a * (b * c) = (a * b)gyr[一个;c, where gyr[a];B]c = gyr(a;b) (c);(ii)所有a;b 2 T, gyr[a;B] = gyr[a] ?b;b]。本文研究了某些陀螺群的正规子陀螺群的结构。
{"title":"Normal subgyrogroups of certain gyrogroups","authors":"Mahdavi Soheila, Ashrafi Ali-Reza, Salahshour Mohammad A.","doi":"10.56415/qrs.v30.09","DOIUrl":"https://doi.org/10.56415/qrs.v30.09","url":null,"abstract":"Suppose that (T;*) is a groupoid with a left identity such that each element a 2 T has a left inverse. Then T is called a gyrogroup if and only if (i) there exists a function gyr : T x T -Aut(T) such that for all a; b; c 2 T, a * (b * c) = (a * b) ? gyr[a; b]c, where gyr[a; b]c = gyr(a; b)(c); and (ii) for all a; b 2 T, gyr[a; b] = gyr[a ? b; b]. In this paper, the structure of normal subgyrogroups of certain gyrogroups are investigated.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43477773","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}
In 1869, Jordan proved that the set T of all finite groups that can be represented as the automorphism group of a tree is containing the trivial group, it is closed under taken the direct product of groups of lower orders in T , and wreath product of a member of T and the symmetric group on n symbols is again an element of T . The aim of this paper is to continue this work and another works by Klavik and Zeman in 2017 to present a class S of finite groups for which the automorphism group of each bicyclic graph is a member of S and this class is minimal with this property.
{"title":"General form of the automorphism group of bicyclic graphs","authors":"Somayeh Madani, A. Ashrafi","doi":"10.56415/qrs.v31.07","DOIUrl":"https://doi.org/10.56415/qrs.v31.07","url":null,"abstract":"In 1869, Jordan proved that the set T of all finite groups that can be represented as the automorphism group of a tree is containing the trivial group, it is closed under taken the direct product of groups of lower orders in T , and wreath product of a member of T and the symmetric group on n symbols is again an element of T . The aim of this paper is to continue this work and another works by Klavik and Zeman in 2017 to present a class S of finite groups for which the automorphism group of each bicyclic graph is a member of S and this class is minimal with this property.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2021-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45510540","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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