Multipotent sets in homogeneous commutative monoids

In this paper, we introduce the concept of k-potent sets in monoids, k ∈ $\mathrm{\mathbb{N}}$, establish their simplest properties, and indicate a class of homogeneous monoids with a set of generating elements. We find simple necessary conditions of the k-potency of a fixed set in such a monoid. For commutative monoids, we establish an isormorphism between them and the monoid ${\mathrm{\mathbb{N}}}_{+}^{\mathfrak{I}}$ with the corresponding label set $\mathfrak{I}$. For commutative homogeneous monoids with sets of generators, we prove necessary and sufficient conditions for the k-potency of their subsets. Finally, we apply this result to the binary Goldbach problem in analytic number theory.