Euler’s totient function applied to complete hypergroups

Abstract

We study the Euler's totient function (called also the Euler's phi function) in the framework of finite complete hypergroups. These are algebraic hypercompositional structures constructed with the help of groups, and endowed with a multivalued operation, called hyperoperation. On them the Euler's phi function is multiplicative and not injective. In the second part of the article we find a relationship between the subhypergroups of a complete hypergroup and the subgroups of the group involved in the construction of the considered complete hypergroup. As sample application of this connection, we state a formula that relates the Euler's totient function defined on a complete hypergroup to the same function applied to its subhypergroups.