Journal of Algebraic Systems
Volume:7 Issue: 2, Winter-Spring 2020

Let G be a group. Recall that the intersection graph of subgroups of G is an undirected graph whose vertex set is the set of all nontrivial subgroups of G and distinct vertices H,K are joined by an edge in this graph if and only if the intersection of H and K is nontrivial. The aim of this article is to investigate the interplay between the group-theoretic properties of a finite group G and the graph-theoretic properties of the complement of the intersection graph of subgroups of G.

The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless Laplacian spectrum.

In this paper we introduce and study a graph on the set of ideals of a commutative ring $R$. The vertices of this graph are non-trivial ideals of $R$ and two distinct ideals $I$ and $J$ are adjacent if and only $IJ=Icap J$. We obtain some properties of this graph and study its relation to the structure of $R$.

In this paper, for any non-empty subset C of a pseudo BCI-algebra X, the concept of p-closure of C, denoted by C(pc), is introduced and some related properties are investigated. Applying this concept, a characterization of the minimal elements of X is given. It is proved that C(pc) is the least closed pseudo BCI-ideal of X containing C and K(X) for any ideal C of X. Finally, by using the concept of p-closure, a closure operator is introduced.

Let M be a right module over a ring R. In this manuscript, we shall study on a special case of F-inverse split modules where F is a fully invariant submodule of M introduced in [12]. We say M is Z 2(M)-inverse split provided f^(-1)(Z2(M)) is a direct summand of M for each endomorphism f of M. We prove that M is Z2(M)-inverse split if and only if M is a direct sum of Z2(M) and a Z2-torsionfree Rickart submodule. It is shown under some assumptions that the class of right perfect rings R for which every right R-module M is Z2(M)-inverse split (Z(M)-inverse split) is precisely that of right GV-rings.

‎The textit{metric dimension} of a connected graph $G$ is the minimum number of vertices in a subset $B$ of $G$ such that all other vertices are uniquely determined by their distances to the vertices in $B$‎. ‎In this case‎, ‎$B$ is called a textit{metric basis} for $G$‎. ‎The textit{basic distance} of a metric two dimensional graph $G$ is the distance between the elements of $B$‎. ‎Giving a characterization for those graphs whose metric dimensions are two‎, ‎we enumerate the number of $n$ vertex metric two dimensional graphs with basic distance 1‎.

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

In this paper, ‎we extend the notion of 2-absorbing ideal on rings to Krasner hyperrings. In fact, we give a characterization of new generalization of prime hyperideals in Krasner hyperrings by introducing 2-absorbing hyperideals‎. ‎We present some illustrative examples. Also, we study fundamental properties of 2-absorbing hyperideals on Krasner hyperrings and investigate some related results.

It is well known that the categories Fuzz of fuzzes and TopFuzz of topological fuzzes are both complete and cocomplete, and some categorical properties of them were introduced by many authors. In this paper, we introduce the structure of equalizers in these categories. In particular, we show that every regular monomorphism is an injective map, but monomorphisms need not be injective, in general.

In this paper, by reviewing the concept of subcovering and semicovering maps, we extend the notion of subcovering map to subsemicovering map. We present some necessary or sufficient conditions for a local homeomorphism to be a subsemicovering map. Moreover, we investigate the relationship between these conditions by some examples. Finally, we give a necessary and sufficient condition for a subsemicovering map to be semicovering.

A {it local antimagic labeling} of a connected graph $G$ with at least three vertices, is a bijection $f:E(G) rightarrow {1,2,ldots , |E(G)|}$ such that for any two adjacent vertices $u$ and $v$ of $G$, the condition $omega _{f}(u) neq omega _{f}(v)$ holds; where $omega _{f}(u)=sum _{xin N(u)} f(xu)$. Assigning $omega _{f}(u)$ to $u$ for each vertex $u$ in $V(G)$, induces naturally a proper vertex coloring of $G$; and $|f|$ denotes the number of colors appearing in this proper vertex coloring. The {it local antimagic chromatic number} of $G$, denoted by $chi _{la}(G)$, is defined as the minimum of $|f|$, where $f$ ranges over all local antimagic labelings of $G$. In this paper, we explicitly construct an infinite class  of connected graphs $G$ such that $chi _{la}(G)$ can be arbitrarily large while $chi _{la}(G vee bar{K_{2}})=3$, where $G vee bar{K_{2}}$ is the join graph of $G$ and the complement graph of $K_{2}$. The aforementioned fact leads us to an infinite class of counterexamples to a result of [Local antimagic vertex  coloring of a graph,  Graphs and Combinatorics 33} (2017), 275-285].

We study primary ideals of the ring $mathcal{R}L$ of real-valued continuous functions on a completely regular frame $L$. We observe that prime ideals and primary ideals coincide in a $P$-frame. It is shown that every primary ideal in $mathcal{R}L$ is contained in a unique maximal ideal, and an ideal $Q$ in $mathcal{R}L$ is primary if and only if $Q capmathcal{R}^*L$ is a primary ideal in $mathcal{R}^*L$. We show that every pseudo-prime (primary) ideal in $mathcal{R}L$ is either an essential ideal or a maximal ideal which is at the same time a minimal prime ideal. Finally, we prove that if $L$ is a connected frame, then the zero ideal in $mathcal{R}L$ is decomposable if and only if $L={bf2}$.

In this paper, we define a structure to obtain exponent matrices of girth-8 QC-LDPC codes with column weight 3. Using the difference matrices introduced by Amirzade et al., we investigate necessary and sufficient conditions which result in a Tanner graph with girth 8. Our proposed method contributes to reduce the search space in recognizing the elements of an exponent matrix. In fact, in this method we only search to obtain one row of an exponent matrix. The other rows are multiplications of that row.

Let R be a commutative Noetherian ring. In this paper we consider some relations between filter regular sequence,regular sequence and system of parameters over R-modules. Also we obtain some new results about cofinitness and cominimaxness of local cohomology modules.

Let ‎$‎mathbb{Z}_p‎$‎ be the finite field of integers modulo ‎$‎p‎$‎, where ‎$‎p>3‎$‎ is a prime integer. This paper presents new constructions of linear codes over ‎$‎mathbb{Z}_p‎$‎‎. Based on our construction, linear codes of length ‎$‎p-1‎$‎, including a wide family of MDS codes, and codes of length ‎$‎(p-1)^2‎$ ‎are ‎constructed.‎ we shall discuss the parameters of the codes defined while describing a generator matrix for the first family.

We show some results about local homology modules and local cohomology modules concerning to being in a serre sub category of the category of R-modules. Also for an ideal I of R we define the concept of CI condition on a serre category, which seems dual to CI condition of Melkerson [1]. As a main result we show that for any minimax R-module M of any serre category S that satisifies CI (CI) condition the local homology module HiI(M) (local cohomology module HIi(M) 2 S) for all i ≥ 0.

In this paper, we study the categorical and algebraic properties, such as limits and colimits of the category Pos-S with respect to order dense embeddings. Injectivity with respect to this class of monomorphisms has been studied by the author and used to obtain information about injectivity relative to regular monomorphisms. Then, we study three dierent kinds of essentiality, usually used in literature, with respect to the class of all order dense embed-dings of S-posets, and investigate their relations to order dense injectivity. We will see, among other things, that although all of these essential extensions are not necessarily equivalent, they behave equivalently with respect to order dense injectivity. More precisely, it is proved that order dense injectivity well behaves regarding these essentialities. Finally, a characterization of these essentialities over pogroups is given.