Algebraic structures and their applications
Year:2016 | Volume:3 | Issue:1|Papers Count:5

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 􀀀  (G), is a graph whose vertices are non trivial proper subgroups of G and two distinct vertices H and K are adjacent if and only if jHj   jKj or jKj   jHj. 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 su cient 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  rst 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) n f(0)g, 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  nite 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 logic entropy for dynamical systems on a D-poset is introduced. Also, the conditional logical entropy is de ned and then some of its properties are studied. The invariance of the 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.