Reza Ameri

In this paper, we investigate the Zariski topology on the prime spectrum of commutative Krasner hyperrings and explore its interplay with the underlying algebraic structure. We characterize the topological properties of the spectrum, such as connectedness, irreducibility, compactness, and separation axioms, and provide necessary and sufficient conditions for each. Notably, we show that the spectrum is irreducible if and only if the nilradical is a prime hyperideal, and it is connected precisely when the hyperring is not a non-trivial product. We also study functorial behavior of the Zariski topology in the category of hyperrings and analyze its correspondence with classical ring theory via the fundamental relation γ∗. Furthermore, we define a topology on the space of prime strongly regular relations and establish a homeomorphism with a subspace of the classical spectrum. These results contribute to the categorical and topological foundations necessary for developing a sheaf-theoretic framework in the context of hyperrings.

In this paper, we introduce and study convex fuzzy cones of hypervector spaces (or simply hyperspaces) as a generalization of both vector spaces and fuzzy vector spaces. We present a new char-acterization of fuzzy subhyperspaces as decompositions of special subhyperspaces. Moreover, we explore convex cones spanned by non-empty subsets of hypervector spaces over the real numbers field. We use fuzzy convex cones to generate fuzzy subhyperspaces from fuzzy subsets. That is, for a given convex fuzzy cone ? on a real hypervector space V, we determine the smallest fuzzy subhyperspace of V containing ? and the largest fuzzy subhyperspace of V contained in ?.

We introduce topological hypervector spaces on a topological field, in the sense of Tallini, and study some basic properties of this hyperspaces. In this regards we study the relationship between the topology on a hypervector spaces and its complete part. In particular we show that if every open subset of a topological hypervector space is a complete part then its fundamental vector space induced is a topological vector space. Finally, we study the quotient space of topological hypervector spaces and the derived topological space of a topological hypervector space with respect its fundamental relegation.

The aim of this paper is the study directly indecomposible multialgebras. In this regards, first the isomorphism theorems and correspondence theorem for multialgebras. Then by applying congruences relation on multialgebras factor multialgebras are constructed and some important properties of them are obtained. In particular, it is shown that every finite multialgebra is isomorphic to a direct products of directly indecomposable of multialgebras. Finally, subdirect products and subdirect irreducible of multialgebras are investigated and Birkoff’s theorem is extended to multialgebras.

We introduce a new regular relation δ on a given group G and show that δ is a congurence relation on G, with respect to module the commutator subgroup of G. Then we show that the effect of this relation on the fundamental relation β is equal to the fundamental relation γ, and we conclude that, if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of δ on ρ, results in a strongly regular relation on H such that its quotient is an abelian group.

The aim of this paper is to study some main properties of convex fuzzy cones of hypervector spaces. In this regard, a characterization of fuzzy subhyperspaces is presented based on ordinary subhyperspaces. Also the convex cone spanned by non-empty subsets of real hypervector spaces is obtained. Moreover, by introducing the notion of fuzzy cone, the smallest fuzzy subhyperspace of V containing µ and the largest fuzzy subhyperspace of V contained in µ is achieved, for a convex fuzzy cone µ of real hypervector space V .

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.