A clonoid based approach to some finiteness results in universal algebraic geometry

We prove that for a finite first order structure $\mathbf{A}$ and a set of first order formulas $\mathrm{\Phi }$ in its language with certain closure properties, the finitary relations on $A$ that are definable via formulas in $\mathrm{\Phi }$ are uniquely determined by those of arity $\mathrm{\mid }A{\mathrm{\mid }}^2$. This yields new proofs for some finiteness results from universal algebraic geometry.