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).
