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.

This paper falls in the area of hypercompositional algebra. In particular it focuses on the class of Krasner hyperrings and it studies the regular local hyperrings. These are Krasner hyperrings $R$ with a unique maximal hyperideal $M$ having the dimension equal to the dimension of the vectorial hyperspace $\frac{M}{M^2}$. The aim of the paper is to show that any regular local hyperring is a hyperdomain. For proving this, we make use of the relationship existing between the dimension of the vectorial hyperspaces related to the hyperring $R$ and to the quotient hyperring $\overline{R}=\frac{R}{⟨a⟩}$, where $a$ is an element in $M\setminus M^2$, and of the regularity of $\overline{R}$.

The aim of this paper is to introduce and study the concept of a hyperideal-based zero-divisor graph associated with a general hyperring. This is a generalized version of the zero-divisor graph associated with a commutative ring. For any general hyperring $R$ having a hyperideal $I$, the $I$-based zero-divisor graph ${\mathrm{\Gamma }}^{(I)}(R)$ associated with $R$ is the simple graph whose vertices are the elements of $R\setminus I$ having their hyperproduct in $I$, and two distinct vertices are joined by an edge when their hyperproduct has a non-empty intersection with $I$. In the first part of the paper, we concentrate on some general properties of this graph related to absorbing elements, while the second part is dedicated to the study of the $I$-based zero-divisor graph associated to the general hyperring ${\mathbb{Z}}_n$ of the integers modulo $n$, when $n=2p^{m}q$, with $p$ and $q$ two different odd primes, and fixing the hyperideal $I$.