Journal of Algebraic Systems
Volume:2 Issue: 2, Winter - Spring 2015

In this paper using strongly duplexes we introduce a new class of (semi)hypergroups. The associated (semi)hypergroup from a strongly duplex is called duplex (semi hypergroup. Two computer programs written in MATLAB show that the two groups $Z_ 2n}$ and $Z_{n}times Z_{2}$ produce a strongly duplex and its associated hypergroup is a complementary feasible hypergroup.

Let $G$ be a simple graph of order $n$ and size $m$. The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial $E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$, where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and $rho(G)$ is the edge covering number of $G$. In this paper we study the edge cover polynomials of cubic graphs of order $10$. We show that all cubic graphs of order $10$ (especially the Petersen graph) are determined uniquely by their edge cover polynomials.

Let $alpha$ be an automorphism of a ring $R$. The authors [On skew inverse Laurent-serieswise Armendariz rings, Comm. Algebra 40(1) (2012) 138-156] applied the concept of Armendariz rings to inverse skew Laurent series rings and introduced skew inverse Laurent-serieswise Armendariz rings. In this article, we study on a special type of these rings and introduce strongly Armendariz rings of inverse skew power series type.
We determine the radicals of the inverse skew Laurent series ring $R((x^{-1};alpha))$, in terms of those of $R$. We also prove that several properties transfer between $R$ and the inverse skew Laurent series extension $R((x^{-1};alpha))$, in case $R$ is a strongly Armendariz ring of inverse skew power series type.

In this paper we use "ring changed'' Gorenstein homological dimensions to define Cohen-Macaulay injective, projective and flat dimensions. For doing this we use the amalgamated duplication of the base ring with semi-dualizing ideals. Among other results, we prove that finiteness of these new dimensions characterizes Cohen-Macaulay rings with dualizing ideals.

In this paper we study the notions of cogenerator and subdirectly irreducible in the category of S-poset. First we give some necessary and sufficient conditions for a cogenerator $S$-posets. Then we see that under some conditions, regular injectivity implies generator and cogenerator. Recalling Birkhoff's Representation Theorem for algebra, we study subdirectly irreducible S-posets and give this theorem for the category of ordered right acts over an ordered monoid. Among other things, we give the relations between cogenerators and subdirectly irreducible S-posets.

Let $G$ be a non-abelian finite group. In this paper, we prove that $Gamma(G)$ is $K_4 -free if and only if $G cong A times P$, where $A$ is an abelian group, $P$ is a $2$ group and $G/Z(G) cong mathbb{ Z}_2 times mathbb{Z}_2$. Also, we show that $Gamma(G)$ is $K_{1,3 }$-free if and only if $G cong {mathbb{S}}_3,~D_8$ or $Q_8$.