Cayley hypergraph over polygroups

Abstract

Comer introduced a class of hypergroups, using the name of polygroups. He emphasized the importance of polygroups, by analyzing them in connections to graphs, relations, Boolean and cylindric algebras. Indeed, polygroups are multi valued systems that satisfy group like axioms. Given a polygroup with a finite generating set, we can form a Cayley hypergraph for that polygroup with respect to that generating set. This helps us to better understand and investigate polygroup structures. More precisely,in this paper, we introduce the construction of Cayley hypergraphs over polygroups, say $CH(\mathbf{P},S)$ such that $\mathbf{P}$ is a polygroup and $\langle S\rangle =P$. We investigate some properties of them. It is well known to give a constructing for building a big polygroup from two small ones. This structure is called extensionof polygroups. In particular, we describe the connection between Cayley hypergraphs over extension of two polygroups and Cartesian product of two Cayley hypergraphs.

Keywords

• cayley hypergraph
• polygroup
• direct hyperproduct
• cartesian product