Gyrogroups abound in group theory. They possess a rich group-like structure that forms a natural generalization of groups. A prominent example is provided by Einstein’s addition law of relativistically admissible velocities. Being nonassociative, it turns out that Einstein addition is a nongroup gyrogroup binary operation. A gyrogroup is a rich structure constituting a non-empty set with a binary operation that obeys an associative-like law called the gyroassociative law. The aim of this article is to present a method of constructing novel gyrogroups of order 2n, n > 3, by a cyclic 2-group that is Z2n.
Given groups (G, ○) and (H, ⋆) with disjoint support, we remark that the isomorphisms (G, ○) → (H, ⋆) are in one-to-one correspondence with certain group operations on G ∪H extending the operation of G and interacting nicely with the structure on H. This correspondence follows from some general properties of groups admitting an index-two subgroup N with an involution x ~∈ N which centralizes N.
We introduce the inclusion prime ideal graph Inp(S) of nontrivial prime ideals of a commutative semigroup S. We characterize a semigroup S for which the graph Inp(S) is null, complete or connected. Then we study various graph parameters, thickness, metric and partition dimension of the inclusion prime ideal graph of the multiplicative semigroup Zn of integers of modulo n. Finally we characterize a ring R for which the graph Inp(R) is null, complete or connected and also some graph theoretic and ring theoretic properties are studied.
We propose several characterizations of solvable ultra-groups and investigate the Jordan-Hölder Theorem and the Zassenhaus Lemma, in ultra-groups. We also define nilpotent ultra-groups by using the center of ultra-groups. Finally, we establish the relation between nilpotent and solvable ultra-groups. Our results aim to serve as a bridge between groups and ultra-groups.
In [6] it was shown that a semiring is completely regular semiring if and only if it is a b-lattice of completely simple semirings. In this paper, we generalize this concept and introduce completely regular R-semiring, completely regular L-semiring and we show that a semiring is completely regular semiring if and only if it is both a completely regular L-semiring and a completely regular R-semiring. Moreover, we show that a semiring is a completely regular R-semiring (completely regular L-semiring) if and only if it is a b-lattice of completely simple R-semirings (completely simple L-semirings).
Full terms that preserve a partition in a finite set are extension of full terms, which can be applied to classify algebras of the same type into subclasses known as the full solid variety preserving a partition. In this paper, two associative binary operations induced by a superassociative superposition defined on the set of full terms preserving a partition are given. As a generalization, sets of such full terms and their binary operations are also discussed.
A loop X is said to be commutative if x2(yz) = (xy)(xz) for all x, y, z 2 X. In the paper, commutative subloops of the Moufang loop of invertible elements in the split octonion algebra over a field of characteristic 2 are described.
Finite fields defined in the form of finite algebras are of significant interest for constructing multivariate-cryptography algorithms with a relatively small size of public key. This application is associated with specifying vector finite fields of dimension m with large number of their modifications for various fixed values of m: A method for parameterized unified generation of multiplication tables of basis vectors is proposed, with the help of which the commutative and associative multiplication operation is specified. The method is represented by a mathematical formula that includes the dimension m and parameters for specifying the distribution of basis vectors and various independent structural constants.
Finite quasigroups and n-quasigroups are currently extensively utilized to implement various cryptographic functions. Cryptographic requirements lead to constraints imposed on quasigroups and n-quasigroups. In particular, V. A. Artamonov proposed using polynomially complete quasigroups. Polynomial completeness can be decided with the help of a criterion of J. Hagemann and C. Herrmann: a quasigroup is polynomially complete if and only if it is simple and non-affine. In our paper we generalize this result to the case of n-quasigroups and give a proof based on I. G. Rosenberg’s description of maximal classes in k-valued logics. We also obtain a completeness criterion and show that completeness is a cryptographically reasonable requirement.
We give a very simple theoretical proof of the fact that the orthomorphism graph of quaternion group Q8 lacks adjacency.
The purpose of this paper is to study O-(dual-) Nijenhuis structures on a Jordan algebra with a representation. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of a Jordan algebra with representation. We introduce the notion of a O-(dual-)Nijenhuis structure on a Jordan algebra with representation. Furthermore, we verify that relative Rota-Baxter operators and O-(dual)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota-Baxter-Nijenhuis structures, r-matrix-Nijenhuis structures, N-structures on a Jordan algebra and we investigate the relation between them.
We prove that if H is a subgroup of a finite solvable group G such that the number of isotopic classes of right transversals of H in G is 1, then H is normal in G.
Let G be a finite group and nse(G) be the set of the number of elements with the same order in G. In this article, we prove that the large Ree groups 2F4(q) with an odd order component prime are uniquely determined by nse(2F4(q)) and their order. As an immediate consequence, we verify Thompson’s problem (1987) for the large Ree groups 2F4(q) with an odd order component prime.
We study the properties of different types of pseudo-ideals of a partially ordered ternary semigroup and prove that the space of all strongly irreducible pseudoideals of a partially ordered ternary semigroup is a compact space.
We characterize the sets of homomorphisms, endomorphisms and automorphisms of n-ary groups with cyclic retracts.
The Ramsey number Rn(3) is the smallest positive integer such that colouring the edges of a complete graph on Rn(3) vertices in n colours forces the appearance of a monochromatic triangle. A lower bound on Rn(3) is obtainable by partitioning the nonidentity elements of a finite group into disjoint union of n symmetric product-free sets. Exact values of Rn(3) are known for n 6 3. The best known lower bound that R4(3) > 51 was given by Chung. In 2006, Kramer gave a proof of over 100 pages that R4(3) 6 62. He then conjectured that R4(3) = 62. We say that the Ramsey number Rn(3) is solvable by group partition means if there is a finite group G such that jGj+1 = Rn(3) and Gnf1g can be partitioned as a union of n symmetric product-free sets. For n 6 3, the Ramsey number Rn(3) is solvable by group partition means. Some authors believe that R4(3) not be solvable by a group partition approach. We prove this here. We also show that any finite group G whose size is divisible by 3 cannot enjoy Gnf1g written as a disjoint union of its symmetric product-free sets. We conclude with a conjecture that R5(3) > 257.
Following the concept of Baer ideals, we define Baer filters and we will make an intensive investigate the basic properties and possible structures of these filters.
Algebraic structures are commonly used as a tool in treatments of various processes. But their exactness reduces the opportunity of their application in nondeterministic environment. On the other hand, probability theory and fuzzy logic do not provide convenient means for expressing the result of combining elements in order to produce new ones. Moreover, these theories are not developed to “measure" algebraic properties. Therefore, we propose a new concept which relies both on universal algebra and probability theory. We introduce probabilistic mappings, and by them we define the notion of a probabilistic algebra. Let A and B be non-empty sets, and let DB be the set of all probability distributions on B. A probabilistic mapping from A to B is a mapping h : A ! DB. Let A be a set, n 2 N, and let An = f(a1; a2; : : : ; an)j ai 2 A; i = 1; 2; : : : ; ng be the n-th power of A. Every probabilistic mapping from An to A is a probabilistic (n-ary) operation on A. A pair (A; F) of a set A and a family F of probabilistic operations on A is called a probabilistic algebra. When F = ff g has one binary operation, then the probabilistic algebra (A; f) is a probabilistic groupoid. “Ordinary" groupoids are just a special type of probabilistic ones. Basic properties of probabilistic groupoids and some classes of probabilistic groupoids (with units, commutative, associative, idempotent, with cancellation, with inverses, quasigroups, groups) are treated in this paper. Here we consider only the finite case.
In the two subvarieties of commutative not necessarily associative rings, and 2-divisible not necessarily associative rings, we show that the sixty Bol-Moufang identities determine exactly five subvarieties in each case, and we completely determine all inclusions and necessary counterexamples. The historically important nonassociative rings of octonions and sedenions may be used as counterexamples in our classification. We utilize both associator and linear forms of each Bol-Moufang identity. The results are analogous to the classification of varieties of loops or quasigroups of Bol-Moufang type (Phillips, Vojtěchovský-2005).
In this paper, we consider an R-group where R is a zero-symmetric right nearring. We define generalized essential ideal of an R-group and prove several properties. Further, we extend this notion to obtain a one-one correspondence between s-essential ideals of R-group and those of Mn(R)-group R n .
