Equations over direct powers of algebraic structures in relational languages

For a semigroup $S$ (group $G$) we study relational equations and describe all semigroups $S$ with equationally Noetherian direct powers. It follows that any group $G$ has equationally Noetherian direct powers if we consider $G$ as an algebraic structure of a certain relational language. Further we specify the results as follows: if a direct power of a finite semigroup $S$ is equationally Noetherian, then the minimal ideal $\text{Ker}(S)$ of $S$ is a rectangular band of groups and $\text{Ker}(S)$ coincides with the set of all reducible elements.