In this paper, we present a program to determine the fuzzy grade of a hypergroup and we exemplify for a certain class of hypergroups associated with genetics. In addition, we highlight the differences between fuzzy grade, strong fuzzy grade, and rough fuzzy grade. MATLAB programs are given to compute the fuzzy grade of this hypergroup and particular cases are presented. Moreover, we mention the algorithm steps and a flowchart of Function getfuzzygrade. In addition, we highlight the differences between fuzzy grade, strong fuzzy grade, and rough fuzzy grade. MATLAB programs are given to compute the fuzzy grade of this hypergroup and particular cases are presented. Moreover, we mention the algorithm steps and a flowchart of Function getfuzzygrade.
We deal with energetic subsets and permeable values in Sheffer stroke BE-algebras. We introduce the notions of S-energetic subsets, F-energetic subsets, lower (resp., upper) permeable S-values, and lower (resp., upper) permeable F-values and find examples to explain them. We create S-energy subsets (resp., F-energy subsets) using Sheffer stroke BE-algebra (resp., Sheffer stroke BE-filter), and vice versa. We find conditions for a lower (resp., upper) level set to be an S-energetic subset or an F-energetic subset. We establish a relationship between a lower (resp., upper) permeable S-value and a lower (resp., upper) permeable F-value. We look at conditions for $\jt \in [0,1]$ to be a lower (resp., upper) permeable S-value or a lower (resp., upper) permeable F-value.
In this paper, we define prime decomposition and normal prime decomposition of an element in an L-module M, prove its uniqueness and study its properties. We revisit normal primary decomposition of an element in an L-module M to obtain few results. Finally, we obtain normal prime decomposition for radical of an element in an L-module M.
Finite hyperoperations are naturally represented by Cayley-type tables whose entries are nonempty subsets of the underlying set. In computational classification, enumeration, and database construction, one first needs coarse but effective ways of identifying tables with the same global entry pattern before applying finer structure-preserving equivalences such as hyperisomorphism or hyperisotopy. Motivated by this need, we introduce an \emph{input-square similarity} on hyperoperations: $\ast_1\sim \ast_2$ if there exists a bijection $\phi:H\times H\to H\times H$ such that $\ast_1=\ast_2\circ \phi$, i.e., the table of $\ast$ is relabeled by permuting input pairs. This relation is the orbit equivalence of the natural action of the symmetric group on $H\times H$. We prove that the frequency profile of table entries is a complete and computable invariant for $\sim$, derive an explicit orbit-size formula via stabilizers, and obtain a closed counting formula for the number of similarity classes on an $n$-element set. We also construct canonical representatives useful for reproducible enumeration and filtering. An explicit example shows that hyperassociativity is not preserved under $\sim$, motivating structurally compatible refinements for axiom-sensitive classification.
In recent literature, Hom-groups have been extensively studied as non-associative generalizations of classical group structures. In this work, we establish a connection between Hom-groups and hypergroups by introducing the notion of Hom-hypergroups as a natural extension of classical hypergroup frameworks. This new structure integrates multi-valued hyperoperations with a Hom-type self-map. We begin by establishing the basic definitions, including Hom-hypermagma and Hom-hypergroup, and we illustrate their fundamental properties through several examples. We then focus on defining and studying Hom-regular equivalence relations, which allow us to construct and analyze quotient structures within the Hom-hypergroup framework. Additionally, we examine how these structures behave under homomorphisms and twisting techniques, showing that key algebraic features are preserved. This study contributes to the ongoing effort to generalize classical algebraic systems and opens new avenues for research in Hom-type hyperstructures.
This paper examines the properties and applications of quasi-order Γ-semihypergroups. We begin by analyzing the relationship between Γ-semihypergroups and classical semihypergroups through a specific regular relation, demonstrating how key results in semihypergroups can be extended to their generalized Γ-semihypergroup counterparts. Furthermore, we investigate fundamental properties of quasi-order Γ-semihypergroups and systematically compare them with their associated semihypergroups using this relation. To extend the theoretical framework, we introduce Γ-hyperideals and explore their structural implications. Additionally, by defining an epimorphism on derived soft rough sets, we establish critical connections between these sets and their generalized soft rough Γ- hyperstructures. Finally, we present a practical application of these Γ-hyperstructures: Leveraging a decision-making method based on Zhan’s algorithms and soft rough structures, we determine the nearest quasi-orderΓ-semihypergroup and its corresponding quasi-order semihypergroup.
This paper introduces and investigates the structure of {\it quantum $d$-algebras} ($Q$-$d$-algebras), a non-commutative generalization of d-algebras defined by a pair of coupled binary operations. We give a condition under which the set $\{x\ast 0: x\in D\}$ to becomes a subalgebra of a $d$-algebra $(D;\ast,0)$. We define a new binary operation $\odot$, % for $i\in \{1,2,3\},$ through composition with $x \ast 0$, and study the resulting algebra $(D; \ast, \odot, 0)$. The central concept is that of a {\it zero-antisymmetric} $d$-algebra, where $x \leq y$ and $y \ast 0 \leq x \ast 0$ imply $x = y.$ We prove that if $(D; \ast, 0)$ is a {\it zero-injective}, then $(D; \odot, 0)$ itself forms a $d$-algebra, and if it is a zero-antisymmetric, then $(D; \ast, \odot, 0)$ forms a pseudo d-algebra. This structure is termed a $Q$-$d$-algebra. We provide several examples and counterexamples to illustrate the theory, and explore the relationship between the standard order $\leq$ and the derived order $\leq_{\odot}$. We show that if $(D; \ast, 0)$ is a zero-injective, then $\leq_{\odot}^1$ is antisymmetric, and if it is a zero-injective d-transitive, then $(D;\leq_{\odot}^1)$ is a poset. Finally, we prove that if $(D; \ast, 0)$ is a zero-injective, and the relation $\leq$ is transitive on the set $\{x\ast 0: x\in D\}$, then $(D;\leq_{\odot}^2)$ is a poset. Further, we prove that if $(D;\ast,\odot,0)$ is a $Q$-$d$-algebra, then $(\frac{D}{\leq_{\odot}};\star, \bigodot,D_0)$ is also a $Q$-$d$-algebra where $\leq_{\odot}=\leq_{\odot}^1\cap\leq_{\odot}^2$ and $D_x\bigodot D_y=(D_x\star D_0)\star(D_y\star D_0)$ for any $D_x$, $D_y$ in $\frac{D}{\leq_{\odot}}$.
Multiplicative hyperrings as an significant type of algebraic hyperstructures extend rings such that the addition is an operation and the multiplication is a hyperoperation. In this paper, we aim to present and characterize two new classes of hyperideals in a commutative multiplicative hyperring called square-difference factor absorbing hyperideals and weakly square-difference factor absorbing hyperideals. We show that the class of square-difference factor absorbing hyperideals is a proper subclass of weakly square-difference factor absorbing hyperideals. We present a range of properties and characterizations for these notions, accompanied by relevant examples. Additionally, we examine the behavior of these classes of hyperideals under various hyperring constructions, including homomorphic images, quotient hyperrings, and cartesian product of multiplicative hyperrings.
In this paper, we study the notion of bi-derivation in Krasner hyperring and provide illustrative examples. We establish several functional identities involving bi-derivation that yield conditions under which a Krasner hyperring becomes commutative. Furthermore, we identify specific identities under which a given bi-derivation $D$ vanishes, thereby characterizing situations where $D$ reduces to the zero map.
In this paper, we introduce the notion of intuitionistic fuzzy GE-filter by combining the concepts of GE-algebras and intuitionistic fuzzy sets. We provide a necessary and sufficient condition for an intuitionistic fuzzy set to form an intuitionistic fuzzy GE-filter. This study examines various properties and characterizations of intuitionistic fuzzy GE-filters. In particular, we explore the roles of (ℏF,∈)t,(ðF,∈)s,(ℏF,q)t,(ðF,q)s,(ℏF,∈ ∨q)t, and (ðF,∈∨q)s sets in determining the filter structures within GE-algebras. Examples illustrate the results, and counterexamples clarify the necessity of the conditions. These results not only enhance the theory of GE-algebras but also contribute to the algebraic treatment of uncertainty using intuitionistic fuzzy logic.
This work rigorously develops ϕ-primary subhypermodules within commutative hyperrings, fundamentally generalizing the classical Lasker-Noether primary decomposition. The analysis focuses on four contributions: (i) Introduction of a refined ϕ-primary definition maintaining coherence with classical theory, tailored for multivalued hyperstructures. (ii) Demonstration of unique algebraic phenomena specific to non-singleton hyperstructures, justifying the hyperring perspective. (iii) Development of algebraic/geometric applications, including generalized radical analysis and proofs of decomposition theorems with conditional uniqueness. (iv) Presentation of detailed examples linking the theory to hyperalgebraic geometry and Tropical Mathematics. This research establishes ϕ-primary submodules as a foundational extension of primary decomposition, creating new avenues for abstract algebra research.
In this paper, the notion of nilpotency is introduced in a Balgebra. Furthermore, some main properties of a nilpotent B-algebra are investigated. Next, by defining the concept of theta pair for a maximal subalgebra of a finite B-algebra, we explore several aspects of this notion to obtain some information about solvable and nilpotent B-algebras.
This paper introduces a new hybrid structure called a multipolar $(m, n)$-fuzzy set, which is an extension of multipolar fuzzy sets and $(m, n)$-fuzzy sets. The concept of a multipolar $(m,n)$-fuzzy set has been applied to the study of semihypergroups. In particular, we introduce the notions of multipolar $(m, n)$-fuzzy sub-semihypergroups, multipolar $(m, n)$-fuzzy hyperideals, multipolar $(m, n)$-fuzzy bi-hyperideals, multipolar $(m, n)$-fuzzy quasi-hyperideals, and multipolar $(m, n)$-fuzzy (1, 2)-hyperideals of semihypergroups, and investigate some of their properties. Additionally, we characterised regular semihypergroups in terms of various types of multipolar $(m, n)$-F-hyperideals.
In this paper, we introduced and investigated the notion of powers of an element of a BCI-algebra and we got their basic properties, also some fundamental results concerning this notation are proved. Then we found the main properties of roots on BCI-algebras and the relation between roots and powers. Moreover, we examine the relationship between roots and powers and show that these concepts are unique.
This article investigates derivation theory within the framework of hoop algebras. We introduce and examine f-derivations and (f, g)-derivations, exploring their properties and providing illustrative examples. We demonstrate that, under specific conditions, the sets of these derivations form bounded distributive lattices and bounded ∨-semilattices. Furthermore, we investigate s-derivations, where s is a square root on the hoop algebra. Notably, we prove that under certain conditions, the set of fixed points of the s-derivation coincides with the set of idempotents of the hoop algebra. This work contributes to the understanding of derivation structures in hoop algebras, particularly focusing on the interplay between derivations, square roots, and idempotent elements.
In this paper we introduces internal states on pseudo $L$-algebras. Firstly, we introduce the notions of internal states of type I and type II on pseudo $L$-algebras, and we also investigate the properties of internal states. Nextly, we study state ideals on internal states pseudo $L$-algebra of type I (type II). Let $(L, \sigma)$ be a type I internal state pseudo $KL$- algebra. If $I$ is a state ideal of $(L, \sigma)$, then $\sigma(I)$ is an ideal of $\sigma(L)$. In the end, we investigate relationships between internal states and Bosbach states, Rie\v{c}an states. Let $\sigma$ be an internal state of type I or type II on a bounded pseudo $L$-algebra $L$ preserving $\rightarrow$ and $\rightsquigarrow$ and $s$ is a Bosbach state on $L$, then the mapping $s_{\sigma}: L \rightarrow [0, 1]$ defined by $s_{\sigma}(x) = s(\sigma(x))$ is a Bosbach state on $L$.
The helix-sum and helix-product overcome restrictions which ordinary sum and product of the classical non-square matrices have. They, in fact, are weak hyperstructures and can be used, among the other applications, in the representation theory of Hv-groups, as well. However, we focus on low dimensions and small cardinality of the underline sets. In the study of helix-products, new interesting Hv-groups, appeared.
The number of Hv-structures, defined on a set is extremely big and gives the chance to applied sciences to work in an opposite to usual direction. Thus one asks from applied sciences to give all restrictions they have, because they reduce the number of possible Hv-structures expressing their problem, by a mathematical model. Moreover, the way of treating Hv-structures gives some new proving methods which can be used in applications. This is the main reason that the Hv-structures have so many applications. Here we present an overview on applications of Hv-structures on other sciences as in Hadronic Mechanics, Lie-Santilli admissible, leptons, biology, teaching, education, and mathematics itself, as well.
This paper investigates the properties of prime and maximal filters in hoops, introducing two distinct types of prime filters and analyzing their interrelationships. Furthermore, it investigates the existence of maximal filters with a special focus on bounded hoops. Moreover, this paper presents the spectrum of a hoop, and shows that when it has the topology of the spectrum, it forms a compact topological space T0. It is also shown that the maximal spectrum, regarded as a subspace of the spectrum, constitutes a compact T1-topological space. Finally, using the Boolean center of a hoop, the paper establishes that the clopen subsets of the hoop algebra coincide with the set of closed subsets generated by Boolean elements.
