Algebraic Structures and Their Applications
https://as.yazd.ac.ir/
Algebraic Structures and Their Applicationsendaily1Wed, 01 May 2024 00:00:00 +0430Wed, 01 May 2024 00:00:00 +0430Right-left induced hyperlattices and the genetic code hyperlattices
https://as.yazd.ac.ir/article_3123.html
&lrm;&lrm;&lrm;In this paper first we introduce right(resp&lrm;. &lrm;left) induced hyperlattices and investigate some of their properties&lrm;. &lrm;Especially&lrm; &lrm;a characterization of the smallest strongly regular relation for the class of distributive right/left induced hyperlattice is investigated&lrm;. &lrm;Next we propose and study the generated hyperlattices from hyperlattices&lrm;. &lrm;Finally&lrm;, &lrm;the right induced hyperlattices of two Boolean lattices of four DNA bases and physico-chemical properties of amino acids of four DNA bases are investigated&lrm;.The duals of annihilator conditions for modules
https://as.yazd.ac.ir/article_3139.html
Let $R$ be a commutative ring with identity and let $M$ be an $R$-module&lrm;. &lrm;The purpose of this paper is to introduce and investigate the submodules of an $R$-module $M$ which satisfy the dual of Property $\mathcal{A}$&lrm;, &lrm;the dual of strong Property $\mathcal{A}$&lrm;, &lrm;and the dual of proper strong Property $\mathcal{A}$&lrm;. &lrm;Moreover&lrm;, &lrm;a submodule $N$ of $M$ which satisfy Property $\mathcal{S_J(N)}$ and Property $\mathcal{I^M_J(N)}$ will be introduced and investigated&lrm;.Modular group algebra with upper Lie Nilpotency index $11p-9$
https://as.yazd.ac.ir/article_3140.html
Let $KG$ be the modular group algebra of a group $G$ over a field $K$ of characteristic $p&gt;0$. Recently, we have seen the classification of group algebras $KG$ with upper Lie nilpotency index $t^{L}(KG)$ up to $10p-8$. In this paper, our aim is to classify the modular group algebra $KG$ with upper Lie nilpotency index $11p-9$, for $G'= \gamma_{2}(G)$ as an abelian group.A study on constacyclic codes over the ring $\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$
https://as.yazd.ac.ir/article_3145.html
This paper studies $\lambda$-constacyclic codes and skew $\lambda$-constacyclic codes over the finite commutative non-chain ring $R=\mathbb{Z}_4+u\mathbb{Z}_4+u^2\mathbb{Z}_4$ with $u^3=0$ for $\lambda= (1+2u+2u^2)$ and $(3+2u+2u^2)$. We introduce distinct Gray maps and show that the Gray images of $\lambda$-constacyclic codes are cyclic, quasi-cyclic, and permutation equivalent to quasi-cyclic codes over $\mathbb{Z}_4$. It is also shown that the Gray images of skew $\lambda$-constacyclic codes are quasi-cyclic codes of length $2n$ and index 2 over $\mathbb{Z}_4$. Moreover, the structure of $\lambda$-constacyclic codes of odd length $n$ over the ring $R$ is determined and give some suitable examples.Semihypergroups that every hyperproduct only contains some of the factors
https://as.yazd.ac.ir/article_3150.html
Breakable semihypergroups, defined by a simple property: every non-empty subset of them is a subsemihypergroup. In this paper, we introduce a class of semihypergroups, in which every hyperproduct of $n$ elements is equal to a subset of the factors, called $\pi_n$-semihypergroups. Then, we prove that every semihypergroup of type $\pi_{2k}$, ($k\geq 2$) is breakable and every semihypergroup of type $\pi_{2k+1}$ is of type $\pi_3$. Furthermore, we obtain a decomposition of a semihypergroup of type $\pi_n$ into the cyclic group of order 2 and a breakable semihypergroup. Finally, we give a characterization of semi-symmetric semihypergroups of type $\pi_n$.On the Ree groups $^2{}G_2(q)$ characterized by a size of a conjugacy class
https://as.yazd.ac.ir/article_3151.html
One of the important problem in finite groups theory is group characterization by specific property. Properties, such as element order, the set of element with the same order, etc. In this paper, we prove that Ree group $^2{}G_2(q)$, where $q\pm\sqrt{3q}+1$ is a prime number can be uniquely determined by its order and one conjugacy class size.Non-commutative hypergroupoid obtained from simple graphs
https://as.yazd.ac.ir/article_3163.html
The purpose of this paper is the study of non-weak commutative hypergroups associated with hypergraphs. In this regards, we construct a hyperoperation on the set of vertices of hypergraph and obtain some results and characterizations of them. Moreover, according to this hyperoperation, we investigate conditions under which the hypergroupoid is a join space hypergroup. Finally, we present an application to marketing social network.
&nbsp;Edge geodetic sequence in graphs
https://as.yazd.ac.ir/article_3181.html
In this paper, we introduced the concept of edge geodetic sequences in graph and its generating function. Some general properties satisfied by this concept are studied. It is shown that for every generating function$$ G(x)=\sum_{i=1}^{\infty} {a}^{i-1}{x^{i-1}} \quad a\in N-\left\lbrace 1\right\rbrace,$$there exists a recurrence graph $G$ with edge geodetic decomposition $\pi=\{G_{1},G_{2},\ldots ,G_{n}\ldots\}$.Locally artinian supplemented modules
https://as.yazd.ac.ir/article_3233.html
In this paper, we introduce notions of RLA-local modules and locally artinian supplemented modules which are proper generalizations as notions of strongly local modules and ss-supplemented modules, respectively and we study some properties of these modules. In particular, we give a characterization of semiperfect rings and left perfect rings.&nbsp;On the maximal Randić energy of trees with given diameter
https://as.yazd.ac.ir/article_3234.html
For given integers $n,d$ with $n\geq 5$ and $4\leq d \leq n-1$, let $T^{n}_d$ be the family of all trees of order $n$ and diameter $d$. In this paper, we study trees $T\in T^{n}_d$ with maximal Randić energy. We prove that if $T\in T^{n}_d$ is a tree with maximal Randić&nbsp; energy then $T$ is obtained from a path $P=v_{0}v_{1} \ldots v_{d}$ by adding $ n_i$ path(s) $P_{3}$ to each vertex $v_{i}$, for $i= 2,3,4,\ldots,d-2$, where $n_i\in \{\lceil\frac{n-d+3}{2d-6}\rceil , \lfloor \frac{n-d+3}{2d-6}\rfloor\}$. In particular, we present families of trees satisfying the Gutman-Furtula-Bozkurt Conjecture proposed in [Linear Algebra Appl., 442 (2014), 50--57].Connections between GE algebras and pre-Hilbert algebras
https://as.yazd.ac.ir/article_3247.html
GE algebras (generalized exchange algebras) and pre-Hilbert algebras are a generalization of well-known Hilbert algebras. In the paper, connections between these algebras are studied. In particular, it is proven that pi-BE algebras are equivalent with GE algebras satisfying the exchange property. Some characterizations of transitive GE algebras and exchange pre-Hilbert algebras are given. It is shown that the intersection of classes of GE algebras and pre-Hilbert algebras is the class of transitive GE algebras. Moreover, GE, BE and pre-Hilbert algebras with the antisymmetry property are investigated. It is proven that transitive GE algebras satisfying the property of antisymmetry coincide with Hilbert algebras. Finally, positive implicative GE and pre-Hilbert algebras are considered, their connections with some algebras of logic are presented. In addition, the hiearchies existing between the classes of algebras studied here are shown.SEMI STRONG OUTER MOD SUM CAYLEY GRAPHS
https://as.yazd.ac.ir/article_3296.html
Let $A$ be an abelian group generated by a $2$-element set $S=\{a, b: a^m=b^n=e, m,n\ge 2\}$, where $e$ is the identity element of $A$. Let $\Gamma_{m,n}=Cay_g(A, S)$ be the undirected Cayley graph of $A$ associated with $S$. In this paper, it is shown that $\Gamma_{2k+1,2l+1}$, $\Gamma_{2, 2+l}$ and $\Gamma_{2k+1, 6}$ are Semi Strong Outer Mod Sum Graphs, and $\Gamma_{k, l}$ is Anti-Outer Mod Sum Graph, for every $k,l\in \mathbb{Z}^+$.Congruences in seminearrings and their correspondence with strong ideals
https://as.yazd.ac.ir/article_3353.html
In this paper, we define the notion of strong ideal of a seminearring $S.$ If $S$ is a nearring or a ring then the concept of a strong ideal of $S$ coincides with the usual ideal of $S.$ We show that there is one-one correspondence between strong ideals of $S$ and strong congruences on $S$. Using the concept of strong ideals, we prove classical isomorphism theorems on $S$. We study insertion of factors property and obtain basic results on equisemiprime ideals.On Left Weakly Jointly Prime $(R,S)$-Modules
https://as.yazd.ac.ir/article_3382.html
Let $R$ and $S$ be commutative rings and $M$ an $(R,S)$-module. A proper $(R,S)$-submodule $P$ of $M$ is called left weakly jointly prime if for each $(R,S)$-submodule $N$ of $M$ and elements $a,b$ of $R$ such that $abNS\subseteq P$ implies either $aNS\subseteq P$ or $bNS\subseteq P$. This paper defines left weakly jointly prime $(R,S)$-modules and presents some of their properties. On the other hand, a ring $R$ is called fully prime if each proper ideal of $R$ is prime. We extend this fact to $(R,S)$-modules. An $(R,S)$-module $M$ is called fully left weakly jointly prime if each proper $(R,S)$-submodule of $M$ is left weakly jointly prime. Moreover, we present some properties of fully left weakly jointly prime $(R,S)$-modules. At the end of this paper, we present our main results about the necessary and sufficient conditions for an arbitrary $(R,S)$-module to be fully left weakly jointly prime.&nbsp;$\phi$-$(k,n)$-absorbing (primary) hyperideals in a Krasner $(m,n)$-hyperring
https://as.yazd.ac.ir/article_3414.html
Various expansions of prime hyperideals have been studied in a Krasner $(m,n)$-hyperring $R$. For instance, a proper hyperideal $Q$ of $R$ is called weakly $(k,n)$-absorbing (primary) provided that for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\{0\}$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. In this paper, we aim to extend the notions to the concepts of $\phi$-$(k,n)$-absorbing and $\phi$-$(k,n)$-absorbing primary hyperideals. Assume that $\phi$ is a function from $ \mathcal{HI}(R)$ to $\mathcal{HI}(R) \cup \{\varnothing\}$ such that $\mathcal{HI}(R)$ is the set of hyperideals of $R$ and $k$ is a positive integer. We call a proper hyperideal $Q$ of $R$ a $\phi$-$(k,n)$-absorbing (primary) hyperideal if for $r_1^{kn-k+1} \in R$, $g(r_1^{kn-k+1}) \in Q-\phi(Q)$ implies that there are $(k-1)n-k+2$ of the $r_i^,$s whose $g$-product is in $Q$ $\Bigl ($ $g(r_1^{(k-1)n-k+2}) \in Q$ or a $g$-product of $(k-1)n-k+2$ of $r_i^,$s ,except $g(r_1^{(k-1)n-k+2})$, is in $\boldsymbol{ r}^{(m,n)}(Q)$ $\Bigr )$. Several properties and characterizations of them are presented.Classification of groups whose vanishing \\elements are contained in exactly two conjugacy classes
https://as.yazd.ac.ir/article_3424.html
Let $G$ be a finite group&lrm;. &lrm;We say that an element $g$ in $G$ is a vanishing element if there exists some irreducible character $\chi$ of $G$ such that $\chi(g)=0$&lrm;. Moreover, the conjugacy class of a vanishing element is called a vanishing conjugacy class. &lrm;In this paper&lrm;, &lrm;we classify groups with exactly two vanishing conjugacy classes and show that such groups are either Frobenius or quasi-Frobenius groups.
&nbsp;On intersection minimal ideal graph of a ring
https://as.yazd.ac.ir/article_3427.html
For a ring $R$, the intersection minimal ideal graph, denoted by $ \wedge(R) $, is a simple undirected graph whose vertices are proper non-zero (right) ideals of $R$ and any two distinct vertices $I_{1}$ and $I_{2}$ are adjacent if and only if $ I_{1} \cap I_{2}$ is a minimal ideal of $R$. In this article, we explore connectedness, clique number, split character, planarity, independence number, domination number of $\wedge(R)$.
&nbsp;On commuting automorphisms and commutator polygroups
https://as.yazd.ac.ir/article_3430.html
We introduce the notions of commuting automorphism and commutator polygroups. The basic question that can be arose about the set of all commuting automorphisms is that for the assumed polygroup $ ( P, \cdot ) $, under what conditions the set of all commuting automorphism $ \boldsymbol {A} ( P) $ is a subgroup of $ Aut ( P) $. In this paper basically the answer to this question is investigated for the class of polygroups.Some results on the sum-annihilating essential submodule graph
https://as.yazd.ac.ir/article_3434.html
Consider a commutative ring $R$ with a non-zero identity $1\neq 0$, and let $M$ be a non-zero unitary module over $R$. In this document, our goal is to present the sum-annihilating essential submodule graph $\mathbb{AE}^{0}_{R}(M)$ and its subgraph $\mathbb{AE}^{1}_{R}(M)$ of a module $M$ over a commutative ring $R$ which is described in the following way: The vertex set of graph $\mathbb{AE}^{0}_{R}(M)$ (resp., $\mathbb{AE}^{1}_{R}(M)$) is the collection of all (resp., non-zero proper) annihilating submodules of $M$ and two separate annihilating submodules $N$ and $K$ are connected anytime $N+K$ is essential in $M$. We study and investigate the basic properties of graphs $\mathbb{AE}^{i}_{R}(M)$ ($i=0, 1$) and will present some related results. Additionally, we explore how the properties of graphs interact with the algebraic structures they represent.Special regular clean rings
https://as.yazd.ac.ir/article_3447.html
In this paper we introduce the concepts of special regular clean elements and regular clean decomposition in a ring R. These concepts lead us to the notion of special regular clean ring. We prove that for a special regular clean element $a=e+r \in R$ and unit $u\in R$ then $au$ is a special regular clean if $u$ is an inner inverse of $e$. We establish that an abelian ring $R$ is a special regular clean ring if and only if the twisted power series ring $R[[x,\sigma]]$ is a special regular clean ring. We also study various characterizations of special clean and special regular clean rings.Spectrum and energies of commuting conjugacy class graph of a finite group
https://as.yazd.ac.ir/article_3450.html
In this paper we compute spectrum, Laplacian spectrum, signless Laplacian spectrum and their corresponding energies of commuting conjugacy class graph of the group $G(p, m, n) = \langle x, y : x^{p^m} = y^{p^n} = [x, y]^p = 1, [x, [x, y]] = [y, [x, y]] = 1\rangle$, where $p$ is any prime, $m \geq 1$ and $n \geq 1$. We derive some consequences along with the fact that commuting conjugacy class graph of $G(p, m, n)$ is super integral. We also compare various energies and determine whether commuting conjugacy class graph of $G(p, m, n)$ is hyperenergetic, L-hyperenergetic or Q-hyperenergetic.Studying some semisimple modules via hypergraphs
https://as.yazd.ac.ir/article_3464.html
Recent studies have shown that hypergraphs are useful in solving real-life problems. Hypergraphs have been successfully applied in various fields. Inspiring by the importance, we shall introduce a new hypergraph assigned to a given module. By the way, vertices of this hypergraph (which we call sum hypergraph) are all nontrivial submodules of a module $P$ and a subset $E$ of the vertices is a hyperedge in case the sum of each two elements of $E$ is equal to $P$ and $E$ is maximal with respect to this condition. Some general properties of such hypergraphs are discussed. Semisimple modules with length $2$ are characterized by their corresponding sum hypergraphs. It is shown that the sum hypergraph assigned to a finite module $P$ is connected if and only if $P$ is semisimple.Meet-nonessential graph of an Artinian lattice
https://as.yazd.ac.ir/article_3466.html
Let $L$ be a lattice with $1$. The meet-nonessential graph $\mathbb{MG} (L)$ of $L$ is a graph whose vertices are all nonessential filters of $L$ and two distinct filters $F$ and $G$ are adjacent if and only if $F \wedge G$ is a nonessential filter of $L$. The basic properties and possible structures of the graph $\mathbb{MG}(L)$ are investigated. The clique number, domination number and independence number of $\mathbb{MG}(L)$ and their relations to algebraic properties of $L$ are explored.&nbsp;