https://as.yazd.ac.ir/ Algebraic Structures and Their Applications en daily 1 Sat, 01 Aug 2026 00:00:00 +0430 Sat, 01 Aug 2026 00:00:00 +0430 https://as.yazd.ac.ir/article_3919.html In this paper, we apply the modules theory to $L$-algebras and introduce the concept of an $L$-module. Then we construct $L$-modules by using power sets and De Morgan algebras. Moreover, we investigate some properties of modules such as sub-module and other related results. Finally, we introduce the concepts of multiplication $L$-modules and co-multiplication $L$-modules and we will represent each sub-module of an $L$-module by using them. https://as.yazd.ac.ir/article_3920.html In this study, we explore prime ideals and prime elements within a semi-ring constructed over a nexus. We characterize these elements using panels and quasi-panels. Furthermore, we establish conditions under which a semi-ring associated with a nexus $N$ becomes unitary. The concept of homomorphism for these semi-rings is introduced, and several of their properties are examined. Additionally, by analyzing their characteristics, we demonstrate that a quotient semi-ring can be induced by an ideal of a semi-ring over a nexus, and localization is successfully defined. To illustrate these concepts, we provide specific examples. https://as.yazd.ac.ir/article_3929.html Let $R$ be a commutative ring, $M$ be an $R$-module and $S\subseteq R$ be a multiplicatively closed subset of $R$. The purpose of this paper is to introduce and investigate the concepts of $S$-co-$r$-submodules and $S$-co-$n$-submodules by using the notion of a multiplicatively closed subset of $R$. A non-zero submodule $N$ of $M$ with $Rad(Ann(M))\cap S=\emptyset$ is called an $S$-co-$n$-submodule, if there exists $s\in S$ such that whenever $aN\subseteq K$ and $sa\not \in Rad(Ann(M))$ for some $a\in R$ and a submodule $K$ of $M$, then $sN\subseteq K$. Many properties and examples are given of such submodules. Also, we state the correspondence between $S$-co-$r$-submodules and $S$-co-$n$-submodules.  https://as.yazd.ac.ir/article_4004.html In this paper, we outline a method for constructing linear codes invariant under primitive permutation groups. We will demonstrate that when a group $G$ possesses a trivial Schur multiplier, every binary linear code that admits $G$ as a permutation group can be regarded as a submodule of the permutation module within the primitive action of $G$. As an illustrative example, we select the finite simple group $G = U(3,3)$ and identify the complete set of linear codes derived from its 2-representations.In addition, we use the supports of the codewords to construct certain designs that remain invariant under the action of $U(3,3)$ and establish connections between these designs and the corresponding linear codes. https://as.yazd.ac.ir/article_4027.html Let $G$ be a finite group. A subgroup $H$ of $G$ is called an $\mathcal{H}{-}$subgroup in $G$ if $H^{g}\cap N_{G}(H)\leq H$ for all $g\in G.$ A subgroup $H$ of $G$ is called an $\mathcal{H}{ C-}$subgroup in $G$ if there exists a normal subgroup $T$ of $G$ such that $G=HT$ and $H^{g}\cap N_{T}(H)\leq H$ for all $g\in G.$ In this paper, we give some new criteria for $p-$nilpotency and supersolvability of a group $G$ when certain subgroups of prime power orders of $G$ are $\mathcal{H}{ C-}$subgroups. https://as.yazd.ac.ir/article_4142.html Let $G$ be a finite group, and let $m(G)=\sum_{g \in G}1/o(g)$, where $o(g)$ denotes the order of $g$. In this paper, we determine all groups such that $m(G)<4$. Additionally, for a finite group $G$ of odd order, we investigate the influence of $m(G)$ on supersolvability. Finally, we provide a criterion for $p$-solvability using the function $m(G)$, where $ p \in \{7,11\}$. https://as.yazd.ac.ir/article_4041.html The $\delta$-primary hyperideal is a concept unifing the $n$-ary prime and $n$-ary primary hyperideals under one frame where $\delta$ is a function which assigns to each hyperideal $Q$ of a hyperring $G$ a hyperideal $\delta(Q)$ of the same hyperring with specific properties. In this paper, for a commutative Krasner $(m,n)$-hyperring $(G,h,k)$ with scalar identity $1$, we aim to introduce and study the notion of $(t,n)$-absorbing $\delta$-semiprimary hyperideals which is a more general structure than $\delta$-primary hyperideals. We say that a proper hyperideal $Q$ of $G$ is an $(t,n)$-absorbing $\delta$-semiprimary hyperideal if whenever $k(a_1^{tn-t+1}) \in Q$ for $a_1^{tn-t+1} \in G$, then there exist $(t-1)n-t+2$ of the $a_i^,$s whose $k$-product is in $\delta(Q)$. Furthermore, we extend the concept to weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideals. Several properties and characterizations of these classes of hyperideals are determined. In particular, after defining srongly weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideals, we present the condition in which a weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal is srongly. Moreover, we show that $k(Q^{(tn-t+1)})=0$ where the weakly $(t,n)$-absorbing $\delta$-semiprimary hyperideal $Q$ is not $(t,n)$-absorbing $\delta$-semiprimary. We give a type of Nakayama$^,$s Lemma on a commutative Krasner $(m,n)$-hyperring. Also, we investigate the stability of the concepts under intersection, homomorphism and cartesian product of hyperrings. https://as.yazd.ac.ir/article_4086.html This paper delves into the investigation of ideals and prime ideals within hoops, a distinct class of algebraic structures. It seeks to identify and analyze specific properties of prime ideals, which are essential for comprehending the behavior of hoops. The notion of a local hoop is introduced, accompanied by an examination of its various equivalent definitions, showcasing its importance within the broader framework of hoop theory. A notable result presented is the proof that if a maximal ideal, denoted as $M$, exists in a hoop, the resulting quotient structure is locally finite, shedding light on the constraints and characteristics of these ideals. Furthermore, the paper introduces a topology on the set of all prime ideals associated with a hoop and explores the conditions required for this topology to be Hausdorff, offering new insights into the topological organization of prime ideals. https://as.yazd.ac.ir/article_4087.html A vertex $v$ in a connected graph $G$ strongly distinguishes two edges $e_1$ and $e_2$ if $\mathrm{d}(e_1,v)= \mathrm{d}(e_2,v)+\mathrm{d}(e_1,e_2)$ or $\mathrm{d}(e_2,v)= \mathrm{d}(e_1,v)+\mathrm{d}(e_1,e_2)$, where for an edge $e=xy$, $\mathrm{d}(e,v)=\mathrm{\min}\{\mathrm{d}(x,v),\mathrm{d}(y,v)\}$ and also the distance between two edges $e_1=xy$ and $e_2=uv$ is $\mathrm{d}(e_1,e_2)=\mathrm{\min}\{\mathrm{d}(e_1,u),\mathrm{d}(e_1,v)\}.$ A nonempty set $S \subseteq V(G)$ is a strong edge metric generator of a graph $G$ if for any two distinct edges $e_1,e_2 \in E(G)$, there exists a vertex $s \in S$ such that $s$ strongly distinguishes $e_1$ and $e_2$. A strong edge metric generating set with the smallest number of elements is called a strong edge metric basis of $G$ and the number of elements in a strong edge metric basis is called the strong edge metric dimension of $G$ and it is denoted by $\mathrm{sedim}(G)$. In this paper, we propose this concept as an extension of the strong metric dimension and we determine the strong edge metric dimension of several classes of graphs. Moreover, the $\mathrm{sedim}$ for the $G$-generalized join graph is investigated. Furthermore, we study the strong edge metric dimension of the comaximal graph of the ring of integers modulo $n$. Finally, we pose an integer linear programming model for finding the strong edge metric dimension of graphs. https://as.yazd.ac.ir/article_4088.html This paper explores several fundamental properties and theorems about bipartiteness, completeness, and connectedness of the Euler function graph $G(\phi(n))$. A key focus of the study is the novel connection between the completeness of the Euler function graph $G(\phi(n))$ and the set $\Phi(n)$. By investigating this connection, we establish new results that clarify the interplay between graph completeness and fundamental properties of prime numbers. Finally, we provide certain examples as well as non-examples. https://as.yazd.ac.ir/article_4122.html In this study, we define a module $\mathcal{H}$ to be Goldie CCLS if and only if there exists a direct summand $\mathcal{B}$ of $\mathcal{H}$ such that $\mathcal{L} \cap \mathcal{B}$ is essential in each of $\mathcal{L}$ and $\mathcal{B}$ for each c-closed submodule $\mathcal{L}$ of $\mathcal{H}$. We examine the structural characteristics of Goldie CCLS modules and identify the connections with the other extending generalizations. We discuss the theory of decomposition. Using examples, we derive several essential features and characterizations of Goldie CCLS modules. https://as.yazd.ac.ir/article_4157.html The main objective of this study is to characterize bitonic algebras by means of the Sheffer stroke and to investigate some properties of related structures. It is shown that every Sheffer stroke bitonic algebra is a bitonic algebra, while the converse holds under special conditions. It is also illustrated that every Sheffer stroke bitonic algebra is a SUP-algebra; however, the converse is not true in general. By introducing certain filters of Sheffer stroke bitonic algebras, a congruence relation is defined on these algebraic structures. Moreover, quotient Sheffer stroke bitonic algebras are constructed via this congruence relation. Finally, homomorphisms on Sheffer stroke bitonic algebras are described, and fundamental homomorphism theorems are proved for these structures. https://as.yazd.ac.ir/article_4158.html For a commutative pointed monoid $S$ and a commutative unital ring $k$, let $k[S]$ be the commutative ring consisting of finite $k$-linear sums of non-zero elements of $S$. In this paper, we investigate some properties of the $k$-realization functor from the category $S$-{\bf Act}$_{0}$ of all pointed $S$-acts to the category $k[S]$-{\bf Mod} of all $k[S]$-modules, which is a left adjoint to the forgetful functor. Using this adjunction, we show that $S$-{\bf Act}$_{0}$ has enough injective objects. Finally, we prove that the functor $k[-]$ is faithful but not full. https://as.yazd.ac.ir/article_4193.html A finite Hamiltonian group is a non-abelian group such that all of its subgroups are normal. This paper studies the power graphs of such groups. By computing the degrees of the vertices of the power graph we manage to give a necessary condition that a non-abelian group of even order to be a Hamiltonian group. Moreover, by presenting an infinite class of non-Hamiltonian groups we prove that the given condition is not sufficient. This study results in two infinite classes of non-isomorphic Eulerian graphs of the same vertex set. The proper power graphs of all Hamiltonian groups have been explicitly identified, as well. https://as.yazd.ac.ir/article_4194.html A topological index is a numerical quantity associated with the topological structure of a graph, used to describe certain of its properties. In this paper, we compute several degree-based topological indices of the zero-divisor graph $\Gamma(R)$ when $R$ is the commutative ring $\mathbb{Z}_p \times \mathbb{Z}_{p^2}$ and $\mathbb{Z}_p \times \mathbb{Z}_{2p}$ for a prime $p>2$. For the commutative ring $R$ with the set of zero-divisors $Z(R)$, the simple graph $\Gamma(R)$ contains the vertex set $V\left(\Gamma(R)\right)=Z(R)\setminus \{0\}$ and the edge set $E\left(\Gamma(R)\right)=\{uv | u, v \in Z(R)\setminus \{0\}~\&~uv=vu=0\}$. https://as.yazd.ac.ir/article_4195.html In this paper, we study double cyclic and double constacyclic code over the field $\mathbf{R_{q}^{n}}$ and a non-chain ring $\mathbf{R_{q}^{n}}=\mathbb{F}_{q}^{n}+u\mathbb{F}_{q}^{n}$, $u^{2}=u.$ We determine the generator polynomials and minimum spanning sets for $\mathbf{R_{q}^{n}}$ and the image of double cyclic code over $\mathbf{R_{q}}$ is a double cyclic code over $\mathbf{R_{q}^{n}}.$ Also we discuss the linear complementary dual codes(LCD) over $\mathbf{R_{q}^{n}}$ has the application in a multi secret-sharing scheme. https://as.yazd.ac.ir/article_4197.html In this paper, we construct a categorical framework for the study of generalized crossed hypermodules and their fundamental connection with generalized hypergroup-groupoids. This approach extends the classical theory of crossed modules to generalized hypergroups, where the binary operation may be partially defined or subject to generalized associativity conditions. We introduce the notion of a generalized hypergroup-groupoid, unifying groupoids and generalized hypergroups, and prove a categorical equivalence between the category of generalized crossed hypermodules and that of generalized hypergroup-groupoids with abelian object sets. Beyond its structural significance, this theory provides an algebraic framework for modeling non-classical symmetries and multi-valued interactions. We present applications in (1) algebraic topology, via generalized homotopy groupoids and higher cohomological actions; (2) quantum and statistical algebra, through multi-state symmetries governed by non-invertible operations; (3) automata and computation theory, by encoding nondeterministic transitions within hypergroupoid structures; (4) network and systems theory, using categorical models of multi-agent interaction spaces; and (5) fuzzy and soft algebraic systems, by formalizing uncertainty and graded associativity through generalized hyperoperations. These examples show that generalized crossed hypermodules provide a unifying language linking algebraic topology, categorical algebra, and non-classical structural models, and suggest further developments in higher-dimensional and homotopical algebra. https://as.yazd.ac.ir/article_4226.html In this paper, we construct a novel hyperoperation derived from a ($BCI$-algebra)$BCK$-algebra with condition $(S)$ and present several significant findings. We establish that this construction leads to the formation of commutative hypergroups and join spaces. Furthermore, we demonstrate that when the $BCK$-algebra {satisfies} condition $(S)$, the resulting hyperstructure constitutes $H_b$-monoids. {Moreover}, we show that every proper subhypergroup of this hypergroup {cannot be} closed, invertible, ultraclosed and conjugable. Finally, we have derived all the hypergroups obtained from $BCK$-algebras with condition $(S)$ of order less than 5. https://as.yazd.ac.ir/article_4234.html In this paper, the notion of $*-$symmetric modules is introduced as an extension of $*-$symmetric rings to modules. Specifically, modules whose endomorphism rings satisfy the $*-$symmetric condition are investigated. Furthermore, we demonstrate that every $*-$symmetric module is a symmetric module, exhibits reversible properties, and satisfies the Insertion of Factors Property. In particular, fundamental results concerning $*-$symmetric and symmetric rings are extended to the context of $*-$symmetric modules. https://as.yazd.ac.ir/article_4252.html Consider three graphs $G_1$, $G_2$, and $H$ with orders $n_1,~n_2, n_3,$ and sizes $m_1,m_2,m_3,$ respectively, where the sets of vertices for all three graphs are disjoint. Let $S(G_1, H)$ be the graph obtained from $G_1$ and $H$ in the following way: (1) Delete all the edges of $G_1$ and consider $m_1$ disjoint copies of $H$. (2) Join each vertex of the $i^{th}$ copy of $H$ to the end vertices of the $i^{th}$ edge of $G_1$.Using this construction, the graph $G_1(\vee_H)G_2$ is defined to be the graph obtained from $S(G_1, H)$ by joining every vertex of $G_1$ to every vertex of $G_2$ (see \cite{som1}). In a similar manner, we define $G_1(\Lambda_H)G_2$ to be the graph formed by connecting every vertex in each of the $m_1$ inserted copies of $H$ in $S(G_1, H)$ to all vertices of $G_2$. We refer to $G_1(\vee_H)G_2$ and $G_1(\Lambda_H)G_2$ as the \emph{triple vertex join} and \emph{triple edge join} of $G_1$ and $G_2$ with respect to $H$, respectively.In this article, we determine the generalized adjacency spectra (also known as the $A_\alpha$-spectra) of $G_1(\vee_H)G_2$ when $G_1$ is regular, and of $G_1(\Lambda_H)G_2$ when both $G_1$ and $H$ are regular. Furthermore, we present an application demonstrating the construction of infinitely many $A_\alpha$-cospectral graphs. https://as.yazd.ac.ir/article_4270.html In this paper, we introduce and study the concept of $S$-$F$-small submodules, where $S$ is a multiplicatively closed subset of a ring $R$, and $F$ is a proper submodule of a right $R$-module $M$. This notion generalizes both $S$-small submodules and $F$-small submodules. Moreover, we establish conditions under which a right $R$-module $M$ is an $S$-$F$-small submodule. Finally, we present the relationships among $S$-essential submodules, $S$-small submodules, $S$-$D$-essential submodules, and $S$-$F$-small submodules under special conditions, where $D$ is a nonzero submodule of a right $R$-module $M$. https://as.yazd.ac.ir/article_4274.html We introduce $H$-regular covering spaces, a natural generalization of classical regular covers where normality in the whole fundamental group is replaced by invariance under conjugation by an arbitrary subgroup $H \le \pi_1(X,x_0)$. We give three equivalent characterizations of $H$-regularity and describe the deck transformation group via a normalizer quotient. When \(H=\pi_1(X,x_0)\), \(H\)-regularity reduces to the usual normality condition \(p_*\pi_1(\widetilde X,\tilde x)\trianglelefteq \pi_1(X,x_0)\); hence \(H\)-regular coverings are exactly ordinary regular coverings. In this case the \(H\)-orbit on each fiber is the whole fiber and the deck transformation group specializes to the classical quotient \(\operatorname{Cov}(\widetilde X/X)\cong \pi_1(X,x_0)/p_*\pi_1(\widetilde X,\tilde x)\). https://as.yazd.ac.ir/article_4275.html In this paper, we introduce and investigate the meet-maximal filter graph of a bounded lattice. It is the (undirected) graph with all nontrivial filters as vertices, and two distinct vertices are adjacent if and only if their meet is a maximal filter. The basic properties and possible structures of the meet-maximal filter graph are investigated. Also, connectedness, clique number, split character, planarity and domination number of meet-maximal filter graph and their relations to algebraic properties of lattices are explored. https://as.yazd.ac.ir/article_4298.html Let ${G}$ be a finite non-abelian group. The non-commuting conjugacy class graph (abbreviated as NCCC-graph) of $G$ is a simple undirected graph whose vertex set is the set of conjugacy classes of non-central elements of $G$ and two vertices $x^G$ and $y^G$ are adjacent to each other if $x'$ and $y'$ do not commute for all $x'\in x^G$ and $y'\in y^G$, where $x^G$ is the conjugacy class of $x \in G$. In this paper, we compute the spectrum, Laplacian spectrum, signless Laplacian spectrum and corresponding energies of NCCC-graphs of certain families of finite non-abelian groups. We determine whether these graphs are integral, L-integral and Q-integral. Further, we compare energy, Laplacian energy and signless Laplacian energy; and determine whether these graphs are borderenergetic, L-borderenergetic, Q-borderenergetic, hyperenergetic, L-hyperenergetic or Q-hyperenergetic. https://as.yazd.ac.ir/article_4299.html Let $M$ be an $R$-module and $T$ a fixed submodule of $M$. Then we say that $M$ is a $T$-semisimple module if each submodule of $M$ is a $T$-direct summand of $M$, i.e. for each submodule $N$ of $M$, there exists a submodule $K$ of $M$ such that $M=N+K$ and $N\cap K$ is contained in $T$. We also present the definition of a $T$-simple module in terms of simple modules. It is shown that a module $M$ is $T$-semisimple if and only if $M/T$ is a semisimple module. We prove that any homomorphic image and each submodule of a $T$-semisimple module inherits the property with a suitable submodule. As expected, it is proved that $M$ is $T$-semisimple if and only if $M$ is a sum of $T$-simple modules. https://as.yazd.ac.ir/article_4300.html For a nonzero commutative ring $A$ with identity and a positive integer $n$, let $R = A\times \cdots\times A$ ($n$ times). The total dot product graph of $R$, denoted by $TD(R)$, is an undirected graph with vertex set equals $R\backslash \{(0, \ldots, 0)\}$ and distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y = 0$, where $x\cdot y$ is the normal dot product of $x$ and $y$. Let $Z(R)$ denote the set of all zero-divisors of $R$ and let us denote $Z(R)\backslash \{(0, \ldots, 0)\}$ by $Z(R)^{*}$. The zero-divisor dot product graph of $R$, denoted by $ZD(R)$, is the subgraph of $TD(R)$ induced by $Z(R)^{*}$. The graph $TD(R)$ (respectively, $ZD(R)$) was introduced and investigated by Badawi [{\it Comm. Algebra} {\bf 43}(1) (2015), 43-50]. We denote the complement of $TD(R)$ (respectively, $ZD(R)$) by $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$). In this paper, we determine the girth of $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$). Moreover, we characterize $R$ such that $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) is complemented. Furthermore, if $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) is connected, then we characterize $R$ such that $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) admits a cut edge.   https://as.yazd.ac.ir/article_4301.html We determine all local $\frac12$-derivations of quasi-filiform Lie algebras of maximum length and compute the dimensions of the corresponding spaces. In every case, the space of local $\frac12$-derivations properly contains the space of $\frac12$-derivations. We also construct explicit non-additive 2-local $\frac12$-derivations on each algebra in the classification.