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Let $S \subset \mathbb{Z}^{d}$ be a finitely generated subsemigroup. Let $E$be a product system over $S$. We show that there exists an infinite dimensionalseparable Hilbert space $\mathcal{H}$ and a semigroup $\alpha:=\{\alpha_x\}_{x\in S}$ of unital normal $*$-endomorphisms of $B(\mathcal{H})$ such that $E$ isisomorphic to the product system associated to $\alpha$.

An essential representation for a product system over a finitely generated subsemigroup of $\mathbb{Z}^{d}$