Algebraic Structures and Their Applications
http://as.yazd.ac.ir/
Algebraic Structures and Their Applicationsendaily1Tue, 01 Feb 2022 00:00:00 +0330Tue, 01 Feb 2022 00:00:00 +0330Quaternary codes and a class of 2-designs invariant under the group $A_8$
http://as.yazd.ac.ir/article_2254.html
In this paper, we use the Key-Moori Method 1 and construct a quaternary code $\mathcal{C}_8$ from a primitive representation of the group $PSL_2(9)$ of degree 15. We see that $\mathcal{C}_8$ is a self-orthogonal even code with the automorphism group isomorphic to the alternating group $A_8$. It is shown that by taking the support of any codeword $\omega$ of weight $l$ in $\mathcal{C}_8$ or $\mathcal{C}_8^\bot$, and orbiting it under $A_8$, a 2-$(15,l,\lambda)$ design invariant under the group $A_8$ is obtained, where $\lambda=\binom{l}{2}|\omega^{A_8}|/\binom{15}{2}$. A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers $(A_8)_\omega$ is determined and moreover, primitivity of $A_8$ on each design is examined.Very true GE-algebras
http://as.yazd.ac.ir/article_2302.html
The concept of very true GE-algebra using very true operator is introduced and its properties are studied to expand the scope of research of GE-algebras. The concepts of simple very true GE-algebra and very true GE-filter are introduced. The characterization of simple very true GE-algebra is discussed, and several properties on very true GE-filter are investigated. Using a very true GE-filter, the quotient very true GE-algebra is constructed, and the uniform and topological space are established.An approach to extending modules via homomorphisms
http://as.yazd.ac.ir/article_2304.html
The notion of $\mathcal{K}$-extending modules was defined recently as a proper generalization of both extending modules and Rickart modules. Let $M$ be a right $R$-module and let $S=End_R(M)$. We recall that $M$ is a $\mathcal{K}$-extending module if for every element $\phi\in S$, $Ker\phi$ is essential in a direct summand of $M$. Since a direct sum of $\mathcal{K}$-extending modules is not a $\mathcal{K}$-extending module in general, an open question is to find necessary and sufficient conditions for such a direct sum to be $\mathcal{K}$-extending. In this paper, we give an answer to this question. We show that if $M_i$ is $M_j$-injective for all $i, j\in I =\{1, 2, \dots, n\}$, then $\bigoplus_{i=1}^n M_i$ is a $\mathcal{K}$-extending module if and only if $M_i$ is $M_j$-$\mathcal{K}$-extending for all $i, j \in I$. Other results on $\mathcal{K}$-extending modules and some of their applications are also included.&nbsp;Weakly primary semi-ideals in posets
http://as.yazd.ac.ir/article_2318.html
One of the main goals of science and engineering is to avail human beings cull the maximum propitious decisions. To make these decisions, we need to ken human being's predictions, feasible outcomes of various decisions, and since information is never absolutely precise and accurate, we need to withal information about the degree of certainty. All these types of information will lead to partial orders. A partially ordered set (or poset) theory deals with partial orders and plays a major role in real life. It has wide range of applications in various disciplines such as computer science, engineering, medical field, science, modeling spatial relationship in geographic information systems (GIS), physics and so on. In this paper, we mainly focus on weakly primary semi-ideal of a poset. We introduce the concepts of weakly primary semi-ideal and weakly $Q$-primary semi-ideal for some prime $Q$ of a poset $P$ and characterize weakly primary semi-ideals of $P$ in terms of prime and primary semi-ideals of $P.$ We provide a counter-example for the existence of weakly primary semi-ideal of $P$ which is not a primary semi-ideal of $P.$ We found an equivalent assertion of primary (respy., weakly primary) semi-ideal $r(K)$ for a semi-ideal $K$ of $P.$ Moreover, we introduce the notion of direct product of weakly primary semi-ideal of $P$ and describe its characteristics.Deductive systems of GE-algebras
http://as.yazd.ac.ir/article_2319.html
A new sub-structure called (vivid) deductive system is introduced and their properties are examined. Conditions for a subset to be a deductive system are provided. The notion of upper GE-set is also introduced, and an example to show that any upper GE-set may not be a deductive system are supplied. Conditions for an upper GE-set to be a deductive system are provided. An upper GE-set is used to consider conditions for a subset to be a deductive system. The characterization of deductive system is established, and relationship between deductive system and vivid deductive system are created. Conditions for a deductive system to be a vivid deductive system are given, and the extension property for vivid deductive system is constructed.A variation of $\delta$-lifting and $\delta$-supplemented modules with respect to an equivalence relation
http://as.yazd.ac.ir/article_2334.html
In this paper we introduce Goldie$^{\ast }$-$\delta $-supplemented modules as follows. A module $M$ is called Goldie$^{\ast }$-$\delta $-supplemented (briefly, G$_{\delta }^{\ast }$-supplemented) if there exists a $\delta $-supplement $T$ of $M$ for every submodule $A$ of $M$ such that $A\beta_{\delta }^{\ast }T$. We say that a module $M$ is called Goldie$^{\ast }$-$\delta $-lifting (briefly, G$_{\delta }^{\ast }$-lifting) if there exists a direct summand $D$ of $M$ for every submodule $A$ of $M$ such that $A\beta_{\delta }^{\ast }D$. Note that the last concept given in [4] as a $\delta $-$H$-supplemented module. We present fundamental properties of these modules. We indicate that these modules lie between $\delta $-lifting and $\delta $-supplemented modules. Also we prove that our modules coincide with some variations of $\delta $-supplemented modules for $\delta $-semiperfect modules.On the cofiniteness of local cohomology modules
http://as.yazd.ac.ir/article_2382.html
Let $R$ be a commutative Noetherian ring with identity, $I$ be an ideal of $R$ and $M$ be an $R$-module such that $Ext^j_R(R/I, M)$ is finitely generated for all $j$. We prove that if $\dim H^i_I(M)\leq 1$ for all $i$, then for any $i \geq 0$ and for any submodule $N$ of $H^i_I(M)$ that is either $I$-cofinite or minimax, the $R$-module $H^i_I(M)/N$ is $I$-cofinite. This generalizes the main result of Bahmanpour and Naghipour [8, Theorem 2.6]. As a consequence, the Bass numbers and Betti numbers of $H^i_I (M)$ are finite for all $i \geq 0$. Also, among other things, we show that if either $\dim R/I\leq 2$ or $\dim M\leq 2$, then for each finitely generated $R$-module $N$, the $R$-module $Ext^j_R (N, H^i_I(M))$ is $I$-weakly cofinite, for all $i \geq 0$ and $j\geq 0$. This generalizes [1, Corollary 2.8].Genus of commuting conjugacy class graph of certain finite groups
http://as.yazd.ac.ir/article_2444.html
For a non-abelian group $G$, its commuting conjugacy class graph $\mathcal{CCC}(G)$ is a simple undirected graph whose vertex set is the set of conjugacy classes of the non-central elements of $G$ and two distinct vertices $x^G$ and $y^G$ are adjacent if there exists some elements $x' \in x^G$ and $y' \in y^G$ such that $x'y' = y'x'$. In this paper we compute the genus of $\mathcal{CCC}(G)$ for six well-known classes of non-abelian two-generated groups (viz. $D_{2n}, SD_{8n}, Q_{4m}, V_{8n}, U_{(n, m)}$ and $G(p, m, n)$) and determine whether $\mathcal{CCC}(G)$ for these groups are planar, toroidal, double-toroidal or triple-toroidal.Automata on genetic structure
http://as.yazd.ac.ir/article_2468.html
In this paper, the authors have represented the genetic structures in terms of automata. With the algebraic structure defined on the genetic code authors defined an automaton on those codons as $\Sigma = (C_G, P, A_M, F, G)$ where $P$ is the set of the four bases $A, C, G, U$ as a set of alphabets or inputs, $C_G$ is the set of all 64 codons, obtained from the ordering of the elements of $P$, as the set of states, $A_M$ is the set of the 20 amino acids as the set of outputs that produce during the process. $F$ and $G$ are transition function and output function respectively. Authors observed that $M(\Sigma) = (\lbrace f_a : a \in P \rbrace, \circ)$ defined on the automata $\Sigma$ where $f_a(q) = F(q, a) = qa,\ \ q \in C_G, a \in P$ is a monoid called the syntactic monoid of $\Sigma$, with $f_a \circ f_b = f_{ba}$ $\forall a, b \in P$. Studying the structure defined in terms of automata it is also observed that the algebraic structure $(M(C_G),\ +,\ \cdot)$ forms a Near-Ring with respect to the two operations $' + '$ and $'\cdot '$ where $M(C_G) = \lbrace f \vert f : C_G \rightarrow C_G \rbrace$.Some aspects of unitary addition Cayley graph of Eisensteinintegers modulo $\textit{n}$
http://as.yazd.ac.ir/article_2469.html
The unitary addition Cayley graph $G_n[\omega]$ of Eisenstein integers modulo $n$ has the vertex set $\mathbb{E}_n[\omega]$, the set of Eisenstein integers modulo $n$. Any two vertices $x=a_1+\omega b_1$, $y=a_2+\omega b_2$ of $G_n[\omega]$ are adjacent if and only if $gcd(N(x+y),n)=1$, where $N$ is the norm of any element of $\mathbb{E}_n[\omega]$ given by $N(a+\omega b)=a^2+b^2-ab$. In this paper we obtain some basic graph invariants such as degree of the vertices, number of edges, diameter, girth, clique number and chromatic number of unitary addition Cayley graph of Eisenstein integers modulo $n$. This paper also focuses on determining the independence number of the above mentioned graph.&nbsp;Some aspects of marginal automorphisms of a finite $p$-group
http://as.yazd.ac.ir/article_2538.html
Let $F$ be a free group, $\mathcal{V}$ be a variety of groups defined by the set of laws $V\subseteq F$ and $G$ be a finite $\mathcal{V}$-nilpotent $p$-group. The automorphism $\alpha$ of $G$ is said to be a marginal automorphism (with respect to $V$), if for all $x\in G$, $x^{-1}x^{\alpha}\in V^{\star}(G)$, where $V^{\star}(G)$ denotes the marginal subgroup of $G$. An automorphism $\alpha$ of $G$ is called an IA-automorphism if $x^{-1}x^{\alpha}\in G'$ for each $x\in G$. An automorphism $\alpha$ of $G$ is called a class preserving if for all $x\in G$, there exists an element $g_x\in G$ such that $x^{\alpha}=g_x^{-1}xg_x$. Let $\operatorname{Aut}^{V^{\star}}(G)$, $\operatorname{Aut}^{G'}(G)$ and $\operatorname{Aut}_c(G)$ respectively, denote the group of all marginal automorphisms, IA-automorphisms and class preserving automorphisms of $G$. In this paper, first we give a necessary and sufficient condition on a finite $\mathcal{V}$-nilpotent $p$-group $G$ such that each marginal automorphism of $G$ fixes the center of $G$ element-wise. Then we characterize all finite $\mathcal{V}$-nilpotent $p$-groups $G$ such that $\operatorname{Aut}^{V^{\star}}(G)=\operatorname{Aut}^{G'}(G)$. Finally, we obtain a necessary and sufficient condition for a finite $\mathcal{V}$-nilpotent $p$-group $G$ such that $\operatorname{Aut}^{V^{\star}}(G)=\operatorname{Aut}_c(G)$.$\omega$-filters of distributive lattices
http://as.yazd.ac.ir/article_2553.html
The notion of $\omega$-filters is introduced in distributive lattices and their properties are studied. A set of equivalent conditions is derived for every maximal filter of a distributive lattice to become an $\omega$-filter which leads to a characterization of quasi-complemented lattices. Some sufficient conditions are derived for proper $D$-filters of a distributive lattice to become an $\omega$-filter. Finally, $\omega$-filters of a distributive lattice are characterized with the help of minimal prime $D$-filters.$\Lambda$-Extension of binary matroids
http://as.yazd.ac.ir/article_2608.html
In this paper, we combine two binary operations $\Gamma$-Extension and element splitting under special conditions, to extend binary matroids. For a given binary matroid $M$, we call a matroid obtained in this way a $\Lambda$-Extension of $M$. We note some attractive properties of this matroid operation, particularly constructing a chordal matroid from a chordal binary matroid.GE-derivations
http://as.yazd.ac.ir/article_2631.html
The notions of $\xi$-inside GE-derivation and $\xi$-outside GE-derivation on a GE-algebra are introduced and its properties are investigated. Conditions for a self-map on GE-algebra to be a $\xi$-inside GE-derivation and a $\xi$-outside GE-derivation are provided. The $\xi$-inside GE-derivation or the $\xi$-outside GE-derivation $\varrho$ are used to form two sets $X_{(\varrho = \xi)}$ and ${\rm ker}(\varrho)$, and GE-subalgebra and GE-filter are studied for these two sets.Some separation axioms in topoframes
http://as.yazd.ac.ir/article_2653.html
This paper is about the extension of some classical separation axioms Hausdorffness, regularity and complete regularity to topoframes. We show that they agree with those in frames except perhaps for complete regularity. The interesting results are about complete regularity, in particular when and how these differ from the frame results. These together with the results about B-filters are the focus of the paper.Characterization of monoids by a generalization of weak flatness property
http://as.yazd.ac.ir/article_2655.html
In [On a generalization of weak flatness property, Asian-European Journal of Mathematics, 14(1) (2021)] we introduce a generalization of weak flatness property, called $(WF)'$, and showed that a monoid $S$ is absolutely $(WF)'$ if and only if $S$ is regular and satisfies Conditions $(R_{(WF)'})$ and $(L_{(WF)'})$. In this paper we continue the characterization of monoids by this property of their (finitely generated, (mono)cyclic, Rees factor) right acts. Also we give a classification of monoids for which $(WF)'$ property of their (finitely generated, (mono)cyclic, Rees factor) right acts imply other properties and vise versa. The aim of this paper is to show that the class of absolutely $(WF)'$ monoids and absolutely (weakly) flat monids are coincide.Binary block-codes of $MV$-algebras and Fibonacci sequences
http://as.yazd.ac.ir/article_2671.html
In this paper, the notion of an $M$-function and cut function on a set are introduced and investigated several properties. We use algebraic properties to introduce an algorithm which show that every finite $MV$-algebras and Fibonacci sequences determines a block-code and presented some connections between Fibonacci sequences, $MV$-algebras and binary block-codes. Furthermore, an $MV$-algebra arising from block-codes is established.On the essential $CP$-spaces
http://as.yazd.ac.ir/article_2674.html
Let $C_c(X)$ be the functionally countable subalgebra of $C(X)$. Essential $CP$-spaces are introduced and investigated algebraically and topologically. It is shown that if $X$ is a proper essential $CP$-space, then $mC_c(X)$ is compact if and only if $\{ \eta \}$ is a $G_\delta$, where $\eta$ is the only non $CP$-point of $X$ and $mC_c(X)$ is the space of minimal prime ideals of $C_c(X)$ which are not maximal. Quasi $F_c$-spaces, $c$-basically disconnect spaces, almost $CP$-spaces and almost essential $CP$-spaces are introduced and studied via essential $CP$-spaces. Finally, $C_c(X)$ as a $CSV$-ring where $X$ is an essential $CP$-space is investigated.A new characterization of Projective Special Unitary groups $\mathbf{ U_3 (3^n )}$ by the order of group and the number of elements with the same order
http://as.yazd.ac.ir/article_2675.html
In this paper, we prove that projective special unitary groups $U_3 (3^n)$, where $ 3^{2n}-3^{n}+1$ is a prime number and $3^n\equiv\pm2(\mod 5)$, can be uniquely determined by the order of group and the number of elements with the same order.Modules whose surjective endomorphisms have a $\gamma$-small kernels
http://as.yazd.ac.ir/article_2677.html
In this paper, we introduce a proper generalization of that of Hopfian modules, called $\gamma$-Hopfian modules. A right $R$-module $M$ is said to be $\gamma$-Hopfian, if any surjective endomorphism of $M$ has a $\gamma$-small kernel. Some basic characterizations of $\gamma$-Hopfian modules are proved. We prove that a ring $R$ is semisimple cosingular if and only if every $R$-module is $\gamma$-Hopfian. Further, we prove that the $\gamma$-Hopfian property is preserved under Morita equivalences. Some other properties of $\gamma$-Hopfian modules are also obtained with examples.