The purpose of this paper is the study of direct limit in the category of (m,n)-ary hypermodules over (m,n)-hyperring R. In this regards, we introduce and study R(m,n) − Hmod, the category of R(m,n) − Hmod, and direct limit in this category. In particular, we study a direct limit of morphisms, direct systems of kernels, and cokernels. Finally, we investigate the relationship between the functor home and direct limit and prove that the functor hom preserves direct limit in category R(m,n) − Hmod.

Hypergroups that have at least one identity element and where each element has at least one inverse are called regular hypergroup. In this regards, for a regular hypergroup $H$, it is shown that there exists a correspondence between the set of all strongly regular relations on $H$ and the set of all normal subhypergroups of $H$ containing $S_{\beta}$. More precisely, it has been proven that for every strongly regular relation $\rho$ on $H$, there exists a unique normal subhypergroup of $H$ containing $S_{\beta}$, such that its quotient is a group, isomorphic to $H/\rho$. Furthermore, this correspondence is extended to a lattice isomorphism between them.

If H is a strongly regular hypergroup, we show that the set of regular relations on H and the set of subhypergroups containing 0_H are two lattices that are isomorphic to each other. In the next step, we introduce and study the properties of functors that are constructed by a sequence of strongly regular relations. This helps us to define a specific type of free objects and tensor products on the category of regular hypergroups.

In this paper, we begin by introducing the concept of fuzzy partial hyperalgebra and exploring the relationships between congruence relations and strong congruence relations within this framework. We then construct an embedding of any fuzzy partial hyperalgebra into a fuzzy hyperalgebra, ensuring that all congruence relations on the embedded fuzzy partialhyperalgebra can be simultaneously extended to the corresponding fuzzy hyperalgebra.

For a hypergroup (H,∘) we consider γ^∗, as the smallest equivalence relation on H such that the quotion (H/γ^∗,⊗) is an abelian group. We study some more properties of γ^∗. Initially, it is investigated which subhypergroup the congruence relation modulo is strongly regular on, and its quotient results in an abelian group? This is directly related to the fundamental relation γ^∗, since such subhypergroups must contain S_γ. Then, we examine the functor γ^∗ from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition γ=δ∗β. For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor γ^∗ affects on the stalks of the sheaves of hypergroups.

We introduce and study fuzzy hypercongruence relation on hyperalgebras as an extension of fuzzy congruence relation of algebras and present some important properties and isomorphism theorems. Then by applying the product concept for two fuzzy hypercongruence relations, it is determined that their product also constitutes a fuzzy hypercongruence relation. Finally, fundamental relation on the product of fuzzy subhyperalgebras are investigated.

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.

For a hypergroup $(H,\circ)$ we consider $\gamma^{\ast}$, as the smallest equivalence relation on $H$ such that the quotion $(H/\gamma^{\ast},\tiny{\otimes})$ is an abelian group. We study some more properties of $\gamma^{\ast}$. Initially, it is investigated which subhypergroup the congruence relation modulo is strongly regular on, and its quotient results in an abelian group? This is directly related to the fundamental relation $\gamma^{\ast}$, since such subhypergroups must contain $S_{\gamma}$. Then, we examine the functor $\gamma^{\ast}$ from a categorical perspective and investigate properties such as continuity and cocontinuity concerning it using the decomposition $\gamma=\delta\tiny{\ast}\beta$. For this purpose, we define the reduced words on strongly regular hypergroups. This has a direct application in studying how the functor $\gamma^{\ast}$ affects on the stalks of the sheaves of hypergroups.

In this article, we will study prime spectrum of Krasner hyperrings and Zariski topology on them, which play an important role in algebraic geometry. Then some results about the relationship between the topological properties of Spec(R) and the algebraic properties of the hyperring R will be proved. In the following, by proving that every strongly regular relation on Krasner hyperrings can be considered as a congruence relation, we will define a topology on the set of strongly regular relations, and investigate its relationship with the Zariski topology. In addition, the effect of fundamental relations on the Zariski topology of Krasner hyperrings will also be investigated.

The aim of this paper is the study of ideals of interval BCI-algebras, de-noted by IBCI-algebras, as a generalization of BCI-algebras. In this regards, we introduce interval ideals of anI BCI-algebras and obtain properties of them. Also, we study atoms of interval BCI-algebras. In particular, we give some equivalent conditions to determine interval atoms of an interval BCI-algebra, based on its degenerated elements.

In this paper, we introduce multi-hyperrings and obtain several related results. Also, we study the concept of sub-multi-hyperring and different operations on multi-hyperrings such that intersection, union, direct product, and homomorphism, and investigate their main properties.

This article has been retracted. A retraction notice can be found at https://doi.org/10.3233/JIFS-219433.

The purpose of this paper is to introduce the concept of autonilpotent polygroups and investigate their properties concerning the automorphism of polygroups. To realize the article’s goals, we present the notation of m-very thin polygroups and construct the (non) commutative very thin polygroups on every (infinite) finite non–empty set, where m ∈ N. As a result of the research, is to show that the set of automorphism of some very thin polygroups is equal to the set of automorphism of special groups. The paper includes implications for the development of automorphism of polygroups, and shows that under some conditions very thin polygroups are autonilpotent polygroups and investigates the connection between of autonilpotent polygroups and nilpotent polygroups. The new conception of autonilpotent polygroups was broached for the in this paper the first time.

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 purpose of this paper is to introduce the notion of intersection graphs based on non-trivial hyperideals in general hyperrings. In order to realise the article’s goals, we design general hyperrings on any given ring and compute the set of all hyperideals of related general hyperring. Moreover to establish of connection between of hyperideals as vertices of it’s associated graph, we introduce the concept of absorbing elements. The main method in this research is based on nonempty intersection of hyperideals, so we discuss on divisor of order of general hyperrings. As a result of the research, is constructing of some necessary and sufficient conditions in intersection graphs to be a connected( complete, Hamiltonian, or Eulerian) graph. Also, is presented notations of annihilator and kernel of homomorphisms via absorbing elements and fundamental relations. The paper includes implications for the development of intersection graph, for modelling the complex hypernetworks by absorbing elements, prime divisors and their relations in hyperideals. The new conception of intersection graphs based on hyperideals of general hyperrings was broached for the in this paper first time.

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 strongly regular relation α on a given group G and show that α is a congruence relation on G, with respect to module the commutator subgroup of G. Then we show that the composition of this relation with the fundamental relation β* is equal to the fundamental and γ are is equal to the relation α. and we conclude that if ρ is an arbitrary strongly regular relation on the hypergroup H, then the effect of α on ρ, results in a strongly regular relation such that its quotients is an abelian group.

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.

. This paper extends multirings to a novel concept as general multirings, investigates their properties and presents a special general multirings as notation of (m, n)-potent general multirings. This study analyzes the differences between class of multirings, general multirings and general hyperrings and constructs the class of (in)finite general multirings based on any given nonempty set. In final, we define the concept of hyperideals in general multirings and compare with hyperideals in other similar (hyper)structures.

Graph theoretic techniques have been widely applied to model many types of links in social systems. Also, algebraic hypercompositional structure theory has demonstrated its systematic application in some problems. Influenced by these mathematical notions, a novel semihypergroup-based graph (SBG) of G=H,E is constructed through the fundamental relation γn on H, where semihypergroup H is appointed as the set of vertices and E is addressed as the set of edges on SBG. Indeed, two arbitrary vertices x and y are adjacent if xγny. The connectivity of graph G is characterized by xγ*y, whereby the connected components SBG of G would be exactly the elements of the fundamental group H/γ*. Based on SBG, some fundamental characteristics of the graph such as complete, regular, Eulerian, isomorphism, and Cartesian products are discussed along with illustrative examples to clarify the relevance between semihypergroup H and its corresponding graph. Furthermore, the notions of geometric space, block, polygonal, and connected components are introduced in terms of the developed SBG. To formulate the links among individuals/countries in the wake of the COVID (coronavirus disease) pandemic, a theoretical SBG methodology is presented to analyze and simplify such social systems. Finally, the developed SBG is used to model the trend diffusion of the viral disease COVID-n in social systems (i.e., countries and individuals).