Chemical Examples in Hypergroups

Hypergroups first were introduced by Marty in 1934. Up to now many researchers have been working on this field of modern algebra and developed it. It is purpose of this paper to provide examples of hypergroups associated with chemistry. The examples presented are connected to construction from chain reactions

Fuzzy n-ary polygroups related to fuzzy points

AbstractRecently, fuzzy n-ary sub-polygroups were introduced and studied by Davvaz, Corsini and Leoreanu-Fotea [B. Davvaz, P. Corsini, V. Leoreanu-Fotea, Fuzzy n-ary sub-polygroups, Comput. Math. Appl. 57 (2008) 141–152]. Now, in this paper, the concept of (∈,∈∨q)-fuzzy n-ary sub-polygroups, (∈¯,∈¯∨q¯)-fuzzy n-ary sub-polygroups and fuzzy n-ary sub-polygroup with thresholds of an n-ary polygroup are introduced and some characterizations are described. Also, we give the definition of implication-based fuzzy n-ary sub-polygroups in an n-ary polygroup, in particular, the implication operators in Łukasiewicz system of continuous-valued logic are discussed

Roughness in Fuzzy Cayley Graphs

Rough set theory is a worth noticing approach for inexact and uncertain system modelling. When rough set theory accompanies with fuzzy set theory, which both are a complementary generalization of set theory, they will be attended by potency in theoretical discussions. In this paper a definition for fuzzy Cayley subsets is put forward as well as fuzzy Cayley graphs of fuzzy subsets on groups inspired from the definition of Cayley graphs. We introduce rough approximation of a Cayley graph with respect to a fuzzy normal subgroup. We introduce the approximation rough fuzzy Cayley graphs and fuzzy rough fuzzy Cayley graphs. The last approximation is the mixture of the other approximations. Some theorems and properties are investigated and proved

On central automorphisms of crossed modules

A crossed module $(T,G,\partial)$ consist of a group homomorphism $\partial:T\rightarrow G$ together with an action $(g,t)\rightarrow{}^{\,g}t$ of $G$ on $T$ satisfying $\partial(^{\,g}t)=g\partial(t)g^{-1}$ and \,^{\partial(s)}t=sts^{-1}, for all $g\in G$ and $s,t\in T$. The term crossed module was introduced by J. H. C. Whitehead in his work on combinatorial homotopy theory. Crossed modules and its applications play very important roles in category theory, homotopy theory, homology and cohomology of groups, algebra, K-theory etc. In this paper, we define Adeny-Yen crossed module map and central automorphisms of crossed modules. If $C^*$ is the set of all central automorphisms of crossed module $(T,G,\partial)$ fixing $Z(T,G,\partial)$ element-wise, then we give a necessary and sufficient condition such that $C^*=I_{nn}(T,G,\partial).$ In this case, we prove $Aut_C(T,G,\partial)\cong Hom((T,G,\partial), Z(T,G,\partial))$. Moreover, when $Aut_C(T,G,\partial)\cong Z(I_{nn}(T,G,\partial)))$, we obtain some results in this respect