Journal of Algebraic Structures and Their Applications
Volume:3 Issue: 1, Winter - Spring 2016

One studies the HX-hypergroups, corresponding to the Chinese hypergroups associated with the direct products of some Z/nZ, calculating their fuzzy grades.

The order graph of a group G, denoted by Γ∗(G), is a graph whose vertices are subgroups of G and two distinct vertices H and K are adjacent if and only if |H|∣∣|K| or |K|∣∣|H|.
In this paper, we study the connectivity and diameter of this graph. Also we give a relation between the order graph and prime graph of a group.

In this article, we show the existence of certain exact sequences with respect to two homology theories, called d-homology and extended d-homology. We present sufficient conditions for the existence of long exact extended d- homology sequence. Also we give some illustrative examples.

Let R be a commutative ring with identity and A(R) be the set of ideals of R with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of R, denoted by AGP(R). It is a (undirected) graph with vertices AP(R)=A(R)∩P(R)∖{(0)}, where P(R) is the set of proper principal ideals of R and two distinct vertices I and J are adjacent if and only if IJ=(0). Then, we study some basic properties of AGP(R). For instance, we characterize rings for which AGP(R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of AGP(R). Finally, we compare the principal ideal subgraph AGP(R) and spectrum subgraph AGs(R).

In this paper, a new invariant called {\it logic entropy} for dynamical systems on a D-poset is introduced. Also, the {\it conditional logical entropy} is defined and then some of its properties are studied. The invariance of the {\it logic entropy} of a system under isomorphism is proved. At the end, the notion of an m-generator of a dynamical system is introduced and a version of the Kolmogorov-Sinai theorem is given.