Some algebraic results concerning linear recurrence sequences

We study the set $\mathcal{L}_{F}$ of all $F$-vector spaces $L(P)$ where $P$ is monic and splits over $F$ and $L(Q)$ denotes the set of linear recurrence sequences over $F$ with characteristic polynomial $Q$. We show that $\mathcal{L}_{F}$ can be endowed with two structures of graded commutative semiring. This study allows us to obtain, in compact forms, the polynomial $P,Q\in F[X]$ such that $L(P)=\prod_{i=1}^{m}L(P_{i})$ and $L(Q)=L(P_{1})\ast\cdots\ast L(P_{m})$, where $P_{1}, \ldots, P_{m}$ are any monic polynomials over $F$.