The purpose of this paper is the study of direct limits in category of Krasner (m, n)-hyperrings. In this regards we introduce and study direct limit of a direct system in category (m, n)-hyperrings. Also, we consider fundamental relation , as the smallest equivalence relation on an (m, n)-hyperring R such that the quotient space is an (m, n)-ring, to introdu...

The purpose of this paper is to compute of fundamental relations of hypergroups. In this regards first we study some basic properties of fundamental relation of hypergroups, then we show that any given group is isomorphic to the fundamental group of a nontrivial hypergroup. Finally we study the connections between categories of hypergroups and groups via the fundam...

the concept of algebraic hyperstructures introduced by marty as a generalization of ordinary algebraic structures. in an ordinary algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. the concept of ?-semihyperrings is a generalization of semirings, a generalization of semihyper rings and a generalizat...

Given a group, we construct a fundamental additive functor on its orbit category. We prove that any isomorphism conjecture valid for this fundamental additive functor holds for all additive functors, like K-theory, cyclic homology, topological Hochschild homology, etc. Finally, we reduce this fundamental isomorphism conjecture to K-theoretic ones.

The concept of algebraic hyperstructures introduced by Marty as a generalization of ordinary algebraic structures. In an ordinary algebraic structure, the composition of two elements is an element, while in an algebraic hyperstructure, the composition of two elements is a set. The concept of ?-semihyperrings is a generalization of semirings, a generalization of semihyper‌rings and a generalizat...

An important tool in the hyperstructure theory is the fundamental relation. The factorization of a multialgebra modulo its fundamental relation provides a functor into the category of universal algebras. The question that lead us to the results we will present is whether this functor commutes with the products. 2000 Mathematics Subject Classification: 08A05, 20N20, 08A99.

The main purpose of this paper is to introduce the concept of general Γ-hypermodules as a generalization of Γ-hypermodules, as a generalization of Γ-modules and as a generalization of modules. Then we extended the isomorphism theorems to general Γhypermodules. Also, it is observer that if N is a normal Γ-subhypermodule of Γ-hypermodule M, then, [M : N∗] is an abelian group. Finally, we show tha...

The Seifert-van Kampen Theorem involves the category Top∗ of spaces with base-point, the category Group of groups and the fundamental group functor π1 : Top∗ → Group; the theorem asserts that the functor π1 preserves certain special colimits. The generalisation of this theorem to all dimensions thus requires answers to three immediate questions, namely, what are the appropriate generalisations ...

There are two fundamental obstructions to representing noncommutative rings via sheaves. First, there is no subcanonical coverage on the opposite of the category of rings that includes all covering families in the big Zariski site. Second, there is no contravariant functor F from the category of rings to the category of ringed categories whose composite with the global sections functor is natur...

We introduce and study category of $(m, n)$-ary hypermodules as a generalization of the category of $(m, n)$-modules as well as the category of classical modules. Also, we study various kinds of morphisms. Especially, we characterize monomorphisms and epimorphisms in this category. We will proceed to study the fundamental relation on $(m, n)$-hypermodules, as an important tool in the study of a...