In this paper, we present algorithms to determine the strong fuzzy grade of a hypergroupoid, exemplifying two types of hypergroupoids. Programs involved with GPU computing for calculating the strong fuzzy grade of the hypergroupoids with large cardinality are given, and a comparison of programs between using GPU and CPU is presented. We also give programs for the Corsini method named join2fuzzy and fuzzy2join. All programs are applicable for any hypergroupoid, except the program GenH, with which the hypergroupoid H to be studied is introduced.

This paper introduces and studies quasi sdf-absorbing ideals as a generalization of sdf-absorbing ideals. We investigate the stability of this property under various constructions, including localization, surjective images, Nagata idealizations, and amalgamations. We establish conditions under which the radical of such ideals is prime and discuss a specific class of rings where quasi sdf-absorption implies the sdf-absorbing primary property. The study concludes with a classification of these ideals in Z and examples distinguishing them from related ideal classes.

All rings considered are commutative with identity, and all modules are assumed to be unital. In this paper, we study R-modules in which every quasi-primary submodule is also primary; we refer to such modules as satisfying condition (*). We present several structural properties of these modules and investigate when the direct sum of two modules M1 and M2 inherits condition (*). In addition, we focus on prime ideals P of a ring R with the property that any P-quasi-primary submodule of an R-module M is automatically P-primary. Prime ideals exhibiting this behaviour are introduced as weak sM-prime ideals relative to M. Our results provide a framework for understanding the interaction between the quasi-primary structure of modules and the prime spectrum of the underlying ring.

In this paper, we investigate some properties and characterizations of cor-submodules which is the dual notion of r-submodules. We prove that every nonzero submodule of a finitely generated module is a co-r-submodule. We investigate when a submodule N of an R-module M contains a co-r-submodule. Also, we study the further properties of co-r-Noetherian modules as a generalization of Noetherian modules.

In this paper, we present an algorithm for determining the strong fuzzy grade of a hypergroup, considering a particular class of hypergroups associated with genetics. A study is carried out depending on the cardinality of the initial hypergroup. The last paragraph of this paper contains the MATLAB R2018b program for the fuzzy grade for cardinals 3,4,5,6,8.

This paper aims to introduce a novel idea of possibility multi-fuzzy soft ordered semigroups for ideals and interior ideals. Various results, formulated as theorems based on these concepts, are presented and further validated with suitable examples. This paper also explores the broad applicability of possibility multi-fuzzy soft ordered semigroups in solving modern decision-making problems. Furthermore, this paper explores various classes of ordered semigroups, such as simple, regular, and intra-regular, using this innovative method. Based on these concepts, some important conclusions are drawn with supporting examples. Moreover, it defines the possibility of multi-fuzzy soft ideals for semiprime ordered semigroups.

The concept of the relation β plays a central role in the study of hypercompositional structures. In this paper, we extend the definition of β to the general framework of hypergroupoids and develop an algorithm to compute its elements and its transitive closure β★. We then apply this algorithm to determine the β-class of partial identities in a hypergroup, a process equivalent to computing the heart of the given algebraic structure. Furthermore, we propose a more general algorithm that is also applicable to the case of Hv-groups. By extracting the quotient set with respect to β★ and endowing it with an appropriate group structure, we obtain the so-called fundamental group. The identity of this fundamental group can then be computed directly, yielding the heart of the structure.

The concept of possibility fuzzy soft sets is a step in a new direction towards a soft set approach that can be used to solve decision-making issues. In this piece of research, an innovative and comprehensive conceptual framework for possibility multi-fuzzy soft ordered semigroups by making use of the notions that are associated with possibility multi-fuzzy soft sets as well as ordered semigroups is introduced. Possibility multi-fuzzy soft ordered semigroups mark a newly developed theoretical avenue, and the central aim of this paper is to investigate it. The focus lies on investigating this newly developed theoretical direction, with practical examples drawn from decision-making and diagnosis practices to enhance understanding and appeal to researchers’ interests. We strictly build the notions of possibility multi-fuzzy soft left (right) ideals, as well as l-idealistic and r-idealistic possibility multi-fuzzy soft ordered semigroups. Furthermore, various algebraic operations, such as union, intersection, as well as AND and OR operations are derived, while also providing a comprehensive discussion of their properties. To clarify these innovative ideas, the theoretical constructs are further reinforced with a set of demonstrative examples in order to guarantee deep and improved comprehension of the proposed framework.

Hypergroups represent a generalization of groups, introduced by Marty, that are rich in applications in several sectors of mathematics and in other fields. An important class of hypergroups called join spaces is presented in this paper, along with some connections to lattice theory, in particular, to modular and to distributive lattices. In particular, we study join spaces associated with chains through functions and we analyze when such join spaces are isomorphic. Moreover, a combinatorial problem is presented for a finite context, focusing on calculating the number of isomorphisms classes of join spaces.

In this paper we consider the cases of simple dominance for dihybrid cross, trihybrid cross and the 4- hybrid cross, in order to notice that the distribution of phenotypes forms a lattice. Moreover, the lattice has an interesting property: it is isomorphic with the lattice of the divisors of 2(k), k being a nonzero natural number.

In this paper we analyze strongly 2-absorbing prime ideals (shortly strongly 2-API), strongly 2-absorbing weak prime ideals (shortly strongly 2-AWPI) and 2-absorbing weak prime ideals (shortly 2-AWPI) in a non-commutative ring, which represent generalization of prime ideals (shortly PI) in a non-commutative ring. The relationship between the strongly 2-API and the 2-absorbing prime ideal (shortly 2-API) is examined. We provide examples to illustrate the new concept of strongly m(a1)-system and strongly m(a2)-system as well as the relationships beween them. Let I be an ideal of R and M be a strongly ma1-system such that I boolean AND M = phi. Then there exists a strongly 2-API P of R containing I such that P boolean AND M = phi. We prove that P is a strongly 2-API if and only if A(1)A(2)A(3) subset of P implies that A(1) subset of P or A(2) subset of P or A(3) subset of P for all ideals A(1), A(2) and A(3) of R.

This paper presents a series of important results from the theory of n-hypergroups. Connections with binary relations and with lattices are presented. Special attention is paid to the fundamental relation and to the commutative fundamental relation. In particular, join n-spaces are analyzed.

In this paper we analyse the center and centralizer of an element in the context of reversible regular hypergroups, in order to obtain the class equation in regular reversible hypergroups, by using complete parts. After an introduction in which basic notions and results of hypergroup theory are presented, particularly complete parts, then we give several properties, characterisations and also examples for the center and centralizer of an element for two classes of hypergroups. The next paragraph is dedicated to hypergroups associated with binary relations. We establish a connection between several types of equivalence relations, introduced by J. Jantosciak, such as the operational relation, the inseparability and the essential indistin-guishability and the conjugacy relation for complete hypergroups. Finally, we analyse Rosenberg hypergroup associated with a conjugacy relation.

A new emerging mathematical tool to handle uncertainty is still not a suitable mathematical tool to tackle complicated problems of advanced fields like structural engineering, decision making problems, coding theory, automata theory and economics involving imprecision and fuzzy parameters. The aim of the present paper is to construct a benchmark theory i.e., (is an element of, is an element of boolean OR qk)-fuzzy soft ideal theory in ordered semigroup S. More precisely, (is an element of, is an element of boolean OR qk)-fuzzy soft bi-ideals in an ordered semi-group S are introduced. Various classes such as regular and intraregular ordered semigroups are classified through fuzzy soft bi-ideals of type (is an element of, is an element of boolean OR qk). Ordinary and fuzzy soft bi-ideals of type (is an element of, is an element of boolean OR qk) are connected using level subsets of < f, A >. Finally, upper/lower parts of (is an element of, is an element of boolean OR qk)-fuzzy soft bi-ideals are developed and several results of ordered semigroups based on these newly developed fuzzy soft bi-ideals are presented.

In this paper, we extend the concept of small subhypermodules to alltypes of hypermodules and give nontrivial examples for this concept. As an application, we define and study lifting hypermodules via small subhypermodules.

Alkanes are highly flammable compound at high temperatures. Complete combustion of gaseous alkanes in presence of sufficient oxygen results in the production of steam (H2O(g)), and carbon dioxide (CO2(g)). The reaction is exothermic. In the present paper, this reaction is modelled in the framework of non geometrical join spaces and we investigate its properties. Closed, invertible, ultraclosed and conjugable subhypergroups are characterized in the obtained join space. Also, associated join ternary space is obtained and similar concepts are represented for it.

In this paper, we introduce generalized quadratic forms and hyperconics over quotient hyperfields as a generalization of the notion of conics on fields. Conic curves utilized in cryptosystems; in fact the public key cryptosystem is based on the digital signature schemes (DLP) in conic curve groups. We associate some hyperoperations to hyperconics and investigate their properties. At the end, a collection of canonical hypergroups connected to hyperconics is proposed.

In this paper, we introduce and analyze the concept of “2-geometric space” [Formula: see text] and the notion of [Formula: see text]-part in [Formula: see text]. Then we define 2-strongly transitive geometric spaces and analyze some examples.

The notion of cubic set was introduced in [23] as a generalization of the notion of fuzzy sets and intuitionistic fuzzy sets. In this paper, we initiate a study of cubic sets in left almost semihypergroups. By using the concept of cubic sets, we introduce the notion of cubic sub LA-semihypergroups (hyperideals and bi-hyperideals) and discuss some basic results on cubic sets in LA-semihypergroups.

We investigate here quotient fuzzy hypermodules, constructed by their subfuzzy hypermodules. Moreover, we continue to study of connections between hypermodules and fuzzy hypermodules, and we show that a subhypermodule of an associated hypermodule of a fuzzy hypermodule can be seen as a subfuzzy hypermodule.