Abstract
Let R be a Krasner hyperring. In this paper, we prove a factorization theorem in the category of Krasner R-hypermodules with inclusion single-valued R-homomorphisms as its morphisms. Then, we prove various isomorphism theorems for a smaller category, i.e., the category of Krasner R-hypermodules with strong single-valued R-homomorphisms as its morphisms. In addition, we show that the latter category is balanced. Finally, we prove that for every strong single-valued R-homomorphism f : A → B and a ∈ A , we have K e r ( f ) + a = a + K e r ( f ) = { x ∈ A ∣ f ( x ) = f ( a ) } .
Highlights
Algebraic hyperstructure theory addresses the study of algebraic objects endowed with multivalued operations, which are intended to generalize classical algebraic structures as groups, rings, or modules [1,2,3,4,5]
We start this section with a factorization theorem for an R-homomorphism between R-hypermodules
Krasner hypermodules have been already considered from the standpoint of category theory by various authors in, e.g., [11,12,13,15], focusing on the properties of different types of homomorphisms, notably the so-called inclusion homomorphisms, strong homomorphisms, and weak homomorphisms, according to their behavior with respect to the multivalued addition
Summary
Algebraic hyperstructure theory addresses the study of algebraic objects endowed with multivalued operations, which are intended to generalize classical algebraic structures as groups, rings, or modules [1,2,3,4,5]. In the framework of that theory, hypergroups play a major role. A hypergroup is basically a set endowed by an associative multivalued binary operation, which fulfills an additional condition called reproducibility. Their inspection may reveal complex relationships among algebra, combinatorics, graphs, and numeric sequences [6,7]. Hyperstructures are inherently more complicated and bizarre than their classical counterparts. One of the main research directions in hyperstructure theory consists of identifying a subclass of a rather generic hyperstructure on the basis of a reasonable set of axioms, symmetries, or properties and proceeding with their analysis, in order to construct a theory that is at the same time general, profound, and beautiful
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