Multipotent sets in homogeneous commutative monoids

In this paper, we introduce the concept of $k$-potent sets in monoids, $k\in\mathbb{N}$, 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 $\mathbb{N}_+^{\mathfrak{J}}$ with the corresponding label set $\mathfrak{J}$. 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.