Multipotent sets in uniform commutative monoids and binary Goldbach problem

The concept of $k$-potent sets in monoids, $k\in {\Bbb N}$ is introduced. Their simple properties are found. The class of uniform monoids with generated elements is selected. The simplest necessary conditions are found in order the fixed set in such monoids is the $k$-potent one. It is proved that each commutative uniform monoid with the system of generated elements is isomorphic to the monoid ${\Bbb N}_+^{\frak J}$ with the correspond label set ${\frak J}$. For such monoids the necessary and sufficient conditions of the $k$-potentiality of sets in it are proved. It is proposed the application of such a result to the analysis of the so-called binary Goldbach problem in additive number theory.