Euler's totient function applied to complete hypergroups

Abstract

<abstract><p>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.</p></abstract>

New Aspects in the Theory of Complete Hypergroups

“…Since in a hypergroup we may have or not identities, the role of the order of an element (as in group theory) is taken by the concept of the period of an element, defined as p(a) = min{k ∈ N | a k ⊆ ω H } and then we define [15] the Euler's totient function as In particular, in a complete hypergroup H = g∈G A g with the underlying group G, the period of the element a ∈ A g is the same as the order of g in G and then exp(G) = exp(H) and the Euler's totient function has the form…”