Journal of Algebraic Systems
Volume:9 Issue: 2, Winter-Spring 2022

Let M(X,A,μ) be the ring of real-valued measurable functions on a measurable space (X,A) with measure μ. In this paper, we study the zero-divisor graph of M(X,A,μ), denoted by Γ(M(X,A,μ)). We give the relationships among graph properties of Γ(M(X,A,μ)), ring properties of M(X,A,μ) and measure properties of (X,A,μ). Finally, we investigate the continuity properties of Γ(M(X,A,μ)).

The commuting graph of a finite group GG, C(G)C(G), is a simple graph with vertex set GG in which two vertices xx and yy are adjacent if and only if xy=yxxy=yx. The aim of this paper is to compute the distance Laplacian spectrum and the distance Laplacian energy of the commuting graph of CACA-groups.

In this article the annihilating-ideal graph of the ring C(X) is studied. We have tried to associate the graph properties of AG(X), the ring properties of C(X) and the topological properties of X. It is shown that X has an isolated point if and only if R is a direct summand of C(X) and this happens if and only if AG(X) is not triangulated. Radius, girth, dominating number and clique number of the AG(X) are investigated. It is proved that c(X) <= dt(AG(X)) ,= w(X) and wAG(X) = χAG(X) = c(X).

In this paper, we first provide some counterexamples and derive some new results concerning the usual and singular spec- trum of an element in continuous inverse algebras. Then we continue our investigation about the characterizations of multiplicative linear maps and their related results in these algebras.

Recently, for each row weight KK and column-weight JJ, 3≤J<K3≤J<K, several classes of (J,K)(J,K) quasi-cyclic (QC) low-density parity-check (LDPC) codes with girth 8 have been constructed explicitly such that their corresponding lower-bounds on the size of circulant permutation matrices (CPMs) have been considered small as possible. In this paper, for J=7J=7, a class of (7,K)(7,K) QC-LDPC codes with girth 8 is generated by an explicit method such that the lower-bounds of the constructed codes remarkably are better than the state-of-the-art bound (K−1)(K2+K)+1(K−1)(K2+K)+1.

The aim of this paper is to extend the notion of geometric groups to geometric hypergroups and to investigate the interaction between algebraic and geometric properties of hypergroups. In this regard, we first define a metric structure on hypergroups via word metric and present some examples on it by using generalized Cayley graphs over hypergroups. Then we study a large scale of geometry with respect to the structure of hypergroups and we prove that metric spaces of finitely generated hypergroups coming from different generating sets are quasi-isometric.

In this paper we present some inequalities for the order, the exponent, and the number of generators of the polynilpotent multiplier, the Baer invariant with respect to the variety of polynilpotent groups of class row (c_1; · · · ; c_t) of a powerful p-group.Our results extend some of Mashakekhy and Maohammadzadeh’s in 2007 to polynilpotent multipliers.

We give a decomposition of total graphs on some finite commutative rings R = Zm, where the set of zero-divisors of R is not an ideal. In particular, we study the total graph T(􀀀(Z2npm))where p is a prime and m and n are positive integers and investigate some graph theoretical properties with some of its fundamental subgraphs.

This paper is devoted to introduce a topology on BL-algebras, makes them semitopological algebras. For any BL-algebra L=(L,∧,∨,∗,\too,0,1)L=(L,∧,∨,∗,\too,0,1), the introduced topology is defined by a distance-like function between elements of LL which is defined by a↔b=(a\toob)∗(b\tooa)a↔b=(a\toob)∗(b\tooa). We will show that when the continuous scale [0,1][0,1] is endowed to be a BL-algebra, then this topology admits some of the most important properties of the metric topology. Finally, we will show that this topology can be examined by a similar topology on dual of BL-algebras as well.

Pseudo-BL algebras are a natural generalization of BL-algebras and of pseudo-MV algebras. In this paper the notions of five different types of derivations on a \pbl\ as generalizations of derivations of a BL-algebra are introduced. Moreover, as an extension of derivations of a \pbl , the notions of (φ,ψ)-derivations are defined on these types. Finally, several related properties are discussed.

In this paper, we investigate some properties of top formal local cohomology FdimM=aM a (M). Among other things, we determine AttR(FdimM=aM a (M)), in the case that FdimM=aM a (M) is an artinian module. Also we show that FdimM=aM a (M) is artinian if and only if it is minimax..

In this paper, we consider the mc-topology on Cc(X), the functionally countable subalgebra of C(X). We show that a Tychonoff space X is countably pseudocompact if and only if the mc-topology and the uc-topology on Cc(X) coincide. It is shown that whenever X is a zero-dimensional space, then Cc(X) is first countable if and only if C(X) with the m-topology is first countable. Also, the set of all zero-divisors of Cc(X) is closed if and only if X is an almost P-space. We show that if X is a strongly zero-dimensional space and U is the set of all units of Cc(X), then the maximal ring of quotients of Cc(U) and Cc(Cc(X)) are isomorphic.