Abstract The balancing and Lucas-balancing numbers are solutions of second order recurrence relations. A linear combination of these numbers can also be obtained as solutions of a fourth order recurrence relation. This recurrence relation can be extended to generalized quaternion algebras. Also, the fourth order recurrence relation has application in coding theory.
{"title":"Fascinating Number Sequences from Fourth Order Difference Equation Via Quaternion Algebras","authors":"A. Patra","doi":"10.7151/dmgaa.1369","DOIUrl":"https://doi.org/10.7151/dmgaa.1369","url":null,"abstract":"Abstract The balancing and Lucas-balancing numbers are solutions of second order recurrence relations. A linear combination of these numbers can also be obtained as solutions of a fourth order recurrence relation. This recurrence relation can be extended to generalized quaternion algebras. Also, the fourth order recurrence relation has application in coding theory.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"229 - 237"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41467743","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}
Abstract The concept of a (strong) set-valued BCK/BCI-morphism in BCK/BCI-algebras is considered, and several properties are investigated. Conditions for a set-valued mapping to be a set-valued BCK/BCI-morphism are given. Using the concept of generalized approximation space, generalized rough subalgebra (ideal) in BCK/BCI-algebras are introduced, and investigate their properties. Using the concept of generalized approximation space and ideal of BCK/bCI-algebra, another type of generalized lower and upper approximations based on the ideal is considered, and then several properties are investigated.
{"title":"Generalized Rough Sets Applied to BCK/BCI-Algebras","authors":"Y. Jun, S. Song, E. Roh","doi":"10.7151/dmgaa.1364","DOIUrl":"https://doi.org/10.7151/dmgaa.1364","url":null,"abstract":"Abstract The concept of a (strong) set-valued BCK/BCI-morphism in BCK/BCI-algebras is considered, and several properties are investigated. Conditions for a set-valued mapping to be a set-valued BCK/BCI-morphism are given. Using the concept of generalized approximation space, generalized rough subalgebra (ideal) in BCK/BCI-algebras are introduced, and investigate their properties. Using the concept of generalized approximation space and ideal of BCK/bCI-algebra, another type of generalized lower and upper approximations based on the ideal is considered, and then several properties are investigated.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"343 - 360"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48897406","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}
Abstract The main purpose of this article is to study ordered semihypergroups in the context of uni-soft quasi-hyperideals. In this article, using the notion of soft-union sets in ordered semihypergroups, we introduce the concept of union-soft (uni-soft) quasi-hyperideal and the related properties are investigated. We prove that every uni-soft left (right) hyperideal is a uni-soft quasi-hyperideal but the converse is not true which is shown with help of an example. We present the characterizations of left (right) simple and completely regular ordered semihypergroups in terms of uni-soft quasi-hyperideals. Furthermore we define semiprime uni-soft quasi-hyperideal and characterize completely regular ordered semihypergroup using this notion.
{"title":"Uni-Soft Quasi-Hyperideals Of Ordered Semihypergroups","authors":"Muhammad Farooq, R. Khan, Asghar Khan, M. Izhar","doi":"10.7151/dmgaa.1363","DOIUrl":"https://doi.org/10.7151/dmgaa.1363","url":null,"abstract":"Abstract The main purpose of this article is to study ordered semihypergroups in the context of uni-soft quasi-hyperideals. In this article, using the notion of soft-union sets in ordered semihypergroups, we introduce the concept of union-soft (uni-soft) quasi-hyperideal and the related properties are investigated. We prove that every uni-soft left (right) hyperideal is a uni-soft quasi-hyperideal but the converse is not true which is shown with help of an example. We present the characterizations of left (right) simple and completely regular ordered semihypergroups in terms of uni-soft quasi-hyperideals. Furthermore we define semiprime uni-soft quasi-hyperideal and characterize completely regular ordered semihypergroup using this notion.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"321 - 342"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42297778","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}
Abstract A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.
{"title":"On eGE-Algebras","authors":"R. Bandaru, N. Rafi, A. Rezaei","doi":"10.7151/dmgaa.1362","DOIUrl":"https://doi.org/10.7151/dmgaa.1362","url":null,"abstract":"Abstract A new algebraic structure was introduced, called an eGE-algebra, which is a generalisation of a GE-algebra and investigated its properties. We explore the definition of filters and the quotient algebra associated with such filters.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"395 - 409"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42572225","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}
Abstract Based on the notion of K* (n, r)-full terms defined by the authors, nd- K* (n, r)-full hypersubstitutions are defined. It turns out that the extension of an nd- K* (n, r)-full hypersubstitution is an endomorphism of the algebra of tree languages of nd- K* (n, r)-full terms.
{"title":"On nd-K* (n, r)-Full Hypersubstitutions","authors":"Khwancheewa Wattanatripop, T. Changphas","doi":"10.7151/dmgaa.1374","DOIUrl":"https://doi.org/10.7151/dmgaa.1374","url":null,"abstract":"Abstract Based on the notion of K* (n, r)-full terms defined by the authors, nd- K* (n, r)-full hypersubstitutions are defined. It turns out that the extension of an nd- K* (n, r)-full hypersubstitution is an endomorphism of the algebra of tree languages of nd- K* (n, r)-full terms.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"213 - 227"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48564611","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}
Abstract In this paper, we introduce Sheffer Stroke UP-algebra (in short, SUP-algebra) and study its properties. We demonstrate that the Cartesian product of two SUP-algebras is a SUP-algebra. After presenting SUP-subalgebras, we define SUP-homomorphisms between SUP-algebras.
{"title":"On Sheffer Stroke Up-Algebras","authors":"T. Oner, T. Katican, A. Saeid","doi":"10.7151/dmgaa.1368","DOIUrl":"https://doi.org/10.7151/dmgaa.1368","url":null,"abstract":"Abstract In this paper, we introduce Sheffer Stroke UP-algebra (in short, SUP-algebra) and study its properties. We demonstrate that the Cartesian product of two SUP-algebras is a SUP-algebra. After presenting SUP-subalgebras, we define SUP-homomorphisms between SUP-algebras.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"381 - 394"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45858822","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}
Abstract The concept of O-filters is introduced in commutative BE-algebras. An equivalent condition is derived for every strong regular filter of a BE-algebra to become an O-filter. The concept of quasi-complemented BE-algebras is introduced and also characterized these classes of BE-algebras in terms of dual annihilators. The concept of strong regular filter is introduced and then quasi-complemented BE-algebras and strong BE-algebras are characterized in terms of strong regular filters and O-filters.
{"title":"Quasi-Complemented BE-Algebras","authors":"V. V. Kumar, M. S. Rao, S. Vali","doi":"10.7151/dmgaa.1365","DOIUrl":"https://doi.org/10.7151/dmgaa.1365","url":null,"abstract":"Abstract The concept of O-filters is introduced in commutative BE-algebras. An equivalent condition is derived for every strong regular filter of a BE-algebra to become an O-filter. The concept of quasi-complemented BE-algebras is introduced and also characterized these classes of BE-algebras in terms of dual annihilators. The concept of strong regular filter is introduced and then quasi-complemented BE-algebras and strong BE-algebras are characterized in terms of strong regular filters and O-filters.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"265 - 282"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47962424","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}
Abstract In this paper we investigate the following result. Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R. If F and G are the two generalized derivation of R such that (F(xy) + G(yx))n − (xy ∓ yx)n = 0, for all x, y ∈ I, then either R is commutative or F (x) = x, G(x) = ∓x for all x ∈ R and n = 1.
{"title":"A Result on Prime Rings with Generalized Derivations","authors":"F. Shujat, Shahoor Khan","doi":"10.7151/dmgaa.1373","DOIUrl":"https://doi.org/10.7151/dmgaa.1373","url":null,"abstract":"Abstract In this paper we investigate the following result. Let R be a prime ring, Q its symmetric Martindale quotient ring, C its extended centroid, I a nonzero ideal of R. If F and G are the two generalized derivation of R such that (F(xy) + G(yx))n − (xy ∓ yx)n = 0, for all x, y ∈ I, then either R is commutative or F (x) = x, G(x) = ∓x for all x ∈ R and n = 1.","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"439 - 446"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41608933","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}
Abstract Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].
{"title":"Classification of Elements in Elliptic Curve Over the Ring 𝔽q[ɛ]","authors":"Bilel Selikh, Douadi Mihoubi, N. Ghadbane","doi":"10.7151/dmgaa.1371","DOIUrl":"https://doi.org/10.7151/dmgaa.1371","url":null,"abstract":"Abstract Let 𝔽q[ɛ] := 𝔽q [X]/(X4 − X3) be a finite quotient ring where ɛ4 = ɛ3, with 𝔽q is a finite field of order q such that q is a power of a prime number p greater than or equal to 5. In this work, we will study the elliptic curve over 𝔽q[ɛ], ɛ4 = ɛ3 of characteristic p ≠ 2, 3 given by homogeneous Weierstrass equation of the form Y 2Z = X3 + aXZ2 + bZ3 where a and b are parameters taken in 𝔽q[ɛ]. Firstly, we study the arithmetic operation of this ring. In addition, we define the elliptic curve Ea,b(𝔽q[ɛ]) and we will show that Eπ0(a),π0(b)(𝔽q) and Eπ1(a),π1(b)(𝔽q) are two elliptic curves over the finite field 𝔽q, such that π0 is a canonical projection and π1 is a sum projection of coordinate of element in 𝔽q[ɛ]. Precisely, we give a classification of elements in elliptic curve over the finite ring 𝔽q[ɛ].","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"283 - 298"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44981146","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}
Abstract The order prime divisor graph 𝒫𝒟(G) of a finite group G is a simple graph whose vertex set is G and two vertices a, b ∈ G are adjacent if and only if either ab = e or o(ab) is some prime number, where e is the identity element of the group G and o(x) denotes the order of an element x ∈ G. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph 𝒫𝒟(G) of a group G. Concentrating on the graph 𝒫𝒟(Dn), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of 𝒫𝒟 (ℤn), 𝒫𝒟 (Sn), 𝒫𝒟 (An).
{"title":"On Order Prime Divisor Graphs of Finite Groups","authors":"M. Sen, S. Maity, Sumanta Das","doi":"10.7151/dmgaa.1372","DOIUrl":"https://doi.org/10.7151/dmgaa.1372","url":null,"abstract":"Abstract The order prime divisor graph 𝒫𝒟(G) of a finite group G is a simple graph whose vertex set is G and two vertices a, b ∈ G are adjacent if and only if either ab = e or o(ab) is some prime number, where e is the identity element of the group G and o(x) denotes the order of an element x ∈ G. In this paper, we establish the necessary and sufficient condition for the completeness of order prime divisor graph 𝒫𝒟(G) of a group G. Concentrating on the graph 𝒫𝒟(Dn), we investigate several properties like degrees, girth, regularity, Eulerianity, Hamiltonicity, planarity etc. We characterize some graph theoretic properties of 𝒫𝒟 (ℤn), 𝒫𝒟 (Sn), 𝒫𝒟 (An).","PeriodicalId":36816,"journal":{"name":"Discussiones Mathematicae - General Algebra and Applications","volume":"41 1","pages":"419 - 437"},"PeriodicalIF":0.0,"publicationDate":"2021-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44456101","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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