Morteza Norouzi

We introduce and study some new directions on soft hypermodules and soft fuzzy hypermodules. In this regard, we apply soft set theory to hypermodules to introduce the classes of soft hypermodules and soft fuzzy hypermodules and obtain their basic properties. In particular, we study the connection between soft hypermodules and soft fuzzy hypermodules by associated (fuzzy) hyperoperations and obtain some related basic results.

With a finite R -module M we associate a hypergraph 𝒞𝒥ℋ _R ( M ) having the set V of vertices being the set of all nontrivial submodules of M . Moreover, a subset E _i of V with at least two elements is a hyperedge if for K, L in E _i there is K ∩ L ≠ = 0 and E _i is maximal with respect to this property. We investigate some general properties of 𝒞𝒥ℋ _R ( M ), providing condition under which 𝒞𝒥ℋ _R ( M ) is connected, and find its diameter. Besides, we study the form of the hypergraph 𝒞𝒥ℋ _R ( M ) when M is semisimple, uniform module and it is a direct sum of its each two nontrivial submodules. Moreover, we characterize finite modules with three nontrivial submodules according to their co-intersection hypergraphs. Finally, we present some illustrative examples for 𝒞𝒥ℋ _R ( M ).

In this paper, the notion of a fuzzy geometric space is introduced. Some structures related to it are studied and connections of them are investigated. By using fuzzy blocks of a fuzzy geometric space, the concept of strongly transitivity of fuzzy geometric spaces is defined and studied. Finally, it is proved that the relation [ ] is transitive on fuzzy hypergroups, by associating strongly transitivity fuzzy geometric spaces to fuzzy hypergroups.

The aim of this paper is to define and study the composition vector spaces as a type of tri-operational algebras. In this regard, by presenting nontrivial examples, it is emphasized that they are a proper generalization of vector spaces and their structure can be characterized by using linear operators. Additionally, some related properties about foundations, composition subspaces and residual elements are investigated. Moreover, it is shown how to endow a vector space with a composition structure by using bijective linear operators. Finally, more properties of the composition vector spaces are presented in connection with linear transformations.

Let R be an (m,n)-hyperring. The Γ∗-relation on R in the sense of Mirvakili and Davvaz [?] is the smallest strong compatible relation such that the quotient R∕Γ∗ is an (m,n)-ring. We use Γ∗-relation to define a fundamental functor, F from the category of (m,n)-hyperrings to the category of (m,n)-rings. Also, the concept of a fundamental (m,n)-ring is introduced and it is shown that every (m,n)-ring is isomorphic to R∕Γ∗ for a nontrivial (m,n)-hyperring R. Moreover, the notions of partitionable and quotientable are introduced and their mutual relationship is investigated. A functor G from the category of classical (m,n)-rings to the category of (m,n)-hyperrings is defined and a natural transformation between the functors F and G is given.