Let G = Cm p o C2 be a generalized dihedral group for an odd prime p and a natural number m, L = M(G; 2) be the RA2 loop obtained from G and F be a finite field of characteristic 2. For the loop algebra F[L], we determine the Jacobson radical J(F[L]) of F[L] and the Wedderburn decomposition of F[L]=J(F[L]). The structure of 1 + J(F[L]) is also determined.
设G = Cm p o C2是奇数素数p和自然数m的广义二面体群，L = m (G;2)为由G和F得到的RA2环，为特征为2的有限域。对于循环代数F[L]，我们确定了F[L]的Jacobson根J(F[L])和F[L]的Wedderburn分解=J(F[L])。确定了1 + J(F[L])的结构。
{"title":"On the finite loop algebra F[M(Cm p x C2, 2)]","authors":"Swati Sidana","doi":"10.56415/qrs.v30.28","DOIUrl":"https://doi.org/10.56415/qrs.v30.28","url":null,"abstract":"Let G = Cm p o C2 be a generalized dihedral group for an odd prime p and a natural number m, L = M(G; 2) be the RA2 loop obtained from G and F be a finite field of characteristic 2. For the loop algebra F[L], we determine the Jacobson radical J(F[L]) of F[L] and the Wedderburn decomposition of F[L]=J(F[L]). The structure of 1 + J(F[L]) is also determined.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44182460","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 purpose of this paper is to provide simple characterizations of the projective objects in the category of finitely supported M-sets. To do so, first, we introduce the notion of zero-retraction monoid and then characterize projective finitely supported M-sets where M contains a zero-retraction monoid.
{"title":"Projective finitely supported M-sets","authors":"Khadijeh Keshvardoost, M. Haddadi","doi":"10.56415/qrs.v30.19","DOIUrl":"https://doi.org/10.56415/qrs.v30.19","url":null,"abstract":"The purpose of this paper is to provide simple characterizations of the projective objects in the category of finitely supported M-sets. To do so, first, we introduce the notion of zero-retraction monoid and then characterize projective finitely supported M-sets where M contains a zero-retraction monoid.","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44600369","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}
It is proved that every involutive equivalential equality algebra (E, ∧, ∼, 1), is an involutive residualted lattice EQ-algebra, which operation ⊗ is defined by x ⊗ y = (x → y 0 ) 0 . Moreover, it is showen that by an involutive residualted lattice EQ-algebra we have an involutive equivalential equality algebra
证明了每一个对合等价等式代数（E，∧，～，1）都是对合剩余格EQ代数，其运算由x定义→ y 0）0。此外，还证明了通过对合剩余格EQ代数，我们有一个对合等价等式代数
{"title":"The relationship between EQ algebras and equality algebras","authors":"A. Paad","doi":"10.56415/qrs.v30.26","DOIUrl":"https://doi.org/10.56415/qrs.v30.26","url":null,"abstract":"It is proved that every involutive equivalential equality algebra (E, ∧, ∼, 1), is an involutive residualted lattice EQ-algebra, which operation ⊗ is defined by x ⊗ y = (x → y 0 ) 0 . Moreover, it is showen that by an involutive residualted lattice EQ-algebra we have an involutive equivalential equality algebra","PeriodicalId":38681,"journal":{"name":"Quasigroups and Related Systems","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49515249","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 objective of this article is to introduce the concept of weak mutiplication semimodule and study several properties which are generalization of corresponding results for multiplication modules. We characterize full prime subsemimodules and full maximal subsemimodules and finally it is shown that in a finitely generated faithful weak multiplication semimodule, every proper full subsemimodule is contained in a maximal full subsemimodule.
{"title":"Weak multiplication semimodule","authors":"S. Maity, Sen Mridul Kanti, Swomin Sabnam","doi":"10.56415/qrs.v30.10","DOIUrl":"https://doi.org/10.56415/qrs.v30.10","url":null,"abstract":"The objective of this article is to introduce the concept of weak mutiplication semimodule and study several properties which are generalization of corresponding results for multiplication modules. We characterize full prime subsemimodules and full maximal subsemimodules and finally it is shown that in a finitely generated faithful weak multiplication semimodule, every proper full subsemimodule is contained in a maximal full subsemimodule.","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":"41670390","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}
To every right or left Bol loop corresponds a middle Bol loop. In this paper, the cores of right Bol loops (RBL) and its corresponding middle Bol loops (MBL) were studied. Their algebraic connections were considered. It was shown that the core of a RBL is elastic and right idempotent. The core of a RBL was found to be alternative (or left idempotent) if and only if its corresponding MBL is right symmetric. If a MBL is right (left) symmetric, then, the core of its corresponding RBL is a medial (semimedial). The core of a middle Bol loop has the left inverse property (automorphic inverse property, right idempotence resp.) if and only if its corresponding RBL has the super anti-automorphic inverse property (automorphic inverse property, exponent 2 resp.). If a RBL is of exponent 2, then, the core of its corresponding MBL is left idempotent. If a RBL is of exponent 2 then: the core of a MBL has the left alternative property (right alternative property) if and only if its corresponding RBL has the cross inverse property (middle symmetry). Some other similar results were derived for RBL of exponent 3.
{"title":"Algebraic connections between right and middle Bol loops and their cores","authors":"B. Osoba, Temitope Gbolahan Jaiyeola","doi":"10.56415/qrs.v30.13","DOIUrl":"https://doi.org/10.56415/qrs.v30.13","url":null,"abstract":"To every right or left Bol loop corresponds a middle Bol loop. In this paper, the cores of right Bol loops (RBL) and its corresponding middle Bol loops (MBL) were studied. Their algebraic connections were considered. It was shown that the core of a RBL is elastic and right idempotent. The core of a RBL was found to be alternative (or left idempotent) if and only if its corresponding MBL is right symmetric. If a MBL is right (left) symmetric, then, the core of its corresponding RBL is a medial (semimedial). The core of a middle Bol loop has the left inverse property (automorphic inverse property, right idempotence resp.) if and only if its corresponding RBL has the super anti-automorphic inverse property (automorphic inverse property, exponent 2 resp.). If a RBL is of exponent 2, then, the core of its corresponding MBL is left idempotent. If a RBL is of exponent 2 then: the core of a MBL has the left alternative property (right alternative property) if and only if its corresponding RBL has the cross inverse property (middle symmetry). Some other similar results were derived for RBL of exponent 3.","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":"45202367","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 S be a regular semigroup, I(S) be the set of ideals of S and M be a subset of I(S). In this paper, we introduce an undirected Cayley graph of S, denoted by Гs,m with elements of I(S) as the vertex set, and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of M such that IK = J or JK = I. We study some basic properties of the graph Гs,m such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Гs,m.
{"title":"The Cayley sum graph of ideals of a semigroup","authors":"Afkhamizadeh Mojgan, Hassankhani Mehdi, Khashyarmanesh Kazem","doi":"10.56415/qrs.v30.01","DOIUrl":"https://doi.org/10.56415/qrs.v30.01","url":null,"abstract":"Let S be a regular semigroup, I(S) be the set of ideals of S and M be a subset of I(S). In this paper, we introduce an undirected Cayley graph of S, denoted by Гs,m with elements of I(S) as the vertex set, and, for two distinct vertices I and J, I is adjacent to J if and only if there is an element K of M such that IK = J or JK = I. We study some basic properties of the graph Гs,m such as connectivity, girth and clique number. Moreover, we investigate the planarity, outerplanarity and ring graph of Гs,m.","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":"45638993","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}
Khaki Zohre, Saan Hossein Mohammadzadeh, Nouri Leila
In this paper first we recall condition (PE) and then will give general properties and a characterization of monoids for which all right acts satisfy this condition. Finally, we give a characterization of monoids, by comparing this property of their acts with some others.
{"title":"Characterization of monoids by condition (P_E)","authors":"Khaki Zohre, Saan Hossein Mohammadzadeh, Nouri Leila","doi":"10.56415/qrs.v30.07","DOIUrl":"https://doi.org/10.56415/qrs.v30.07","url":null,"abstract":"In this paper first we recall condition (PE) and then will give general properties and a characterization of monoids for which all right acts satisfy this condition. Finally, we give a characterization of monoids, by comparing this property of their acts with some others.","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":"43943874","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. S. Ebrahimi, Khoramdel Mehdi, Pish Hesari Saboura Dolati
Let L be a lattice with the greatest element 1. Following the concept of Radsupplemented modules, we define Rad-supplemented filters and we will make an intensive investigate the basic properties and possible structures of these filters.
{"title":"Rad-supplemented property in the lattices","authors":"A. S. Ebrahimi, Khoramdel Mehdi, Pish Hesari Saboura Dolati","doi":"10.56415/qrs.v30.05","DOIUrl":"https://doi.org/10.56415/qrs.v30.05","url":null,"abstract":"Let L be a lattice with the greatest element 1. Following the concept of Radsupplemented modules, we define Rad-supplemented filters and we will make an intensive investigate the basic properties and possible structures of these filters.","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":"45225374","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 semilattice equivalence relations play an important role in investigating the structural properties of ordered semihypergroups. Such relations can be expressed in terms of hyperfilters. There are two concepts of (ordered) hyperfilters of (ordered) semihypergroups which were introduced by Tang et al. [16] and Kehayopulu[9]. In this paper, we prove that those two concepts coincide and characterize the least semilattice equivalence relations on ordered semihypergroups. Furthermore, we investigate the relationship between the semilattice equivalence relations and the strongly ordered regular equivalence relations on ordered semihypergroups. Finally, we introduce the concept of p-classes-chain on ordered semihypergroups and give the characterization of the strongly ordered regular equivalence relations via such concept.
{"title":"The ordered semilattice equivalence relations on ordered semihypergroups","authors":"Daengsaen Jukkrit, Leeratanavalee Sorasak","doi":"10.56415/qrs.v30.02","DOIUrl":"https://doi.org/10.56415/qrs.v30.02","url":null,"abstract":"The semilattice equivalence relations play an important role in investigating the structural properties of ordered semihypergroups. Such relations can be expressed in terms of hyperfilters. There are two concepts of (ordered) hyperfilters of (ordered) semihypergroups which were introduced by Tang et al. [16] and Kehayopulu[9]. In this paper, we prove that those two concepts coincide and characterize the least semilattice equivalence relations on ordered semihypergroups. Furthermore, we investigate the relationship between the semilattice equivalence relations and the strongly ordered regular equivalence relations on ordered semihypergroups. Finally, we introduce the concept of p-classes-chain on ordered semihypergroups and give the characterization of the strongly ordered regular equivalence relations via such concept.","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":"42158610","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}
Here we study the principal left k-radicals of a semiring with semilattice additive reduct and characterize the semirings which are disjoint union of principal left k-radicals via the −→ transitive closure l ∞ of the relation −→l on a semiring S, given by for a, b ∈ S, a −→l b ⇔ bn ∈ Sa for some n ∈ N.
本文研究了具有半格加性约的半环的主左k根，并利用半环S上的关系-→l的-→传递闭包l∞刻画了主左k根不相交并的半环，给出了对于a, b∈S，对于n∈n, a -→l b⇔bn∈Sa。
{"title":"Semirings which are union of principal left k-radicals","authors":"T. K. Mondal","doi":"10.56415/qrs.v30.12","DOIUrl":"https://doi.org/10.56415/qrs.v30.12","url":null,"abstract":"Here we study the principal left k-radicals of a semiring with semilattice additive reduct and characterize the semirings which are disjoint union of principal left k-radicals via the −→ transitive closure l ∞ of the relation −→l on a semiring S, given by for a, b ∈ S, a −→l b ⇔ bn ∈ Sa for some n ∈ N.","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":"42888649","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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