Let us call a pair of two real polynomials $(P,Q)$ interlacing if $\deg(P)=\deg(Q)+1$, both polynomials have strictly positive leading coefficients and $P,Q$ have only real roots which interlace weakly, ie. $$\rho_1\leq \rho'_1\leq \rho_2\leq \dots\leq \rho'_{n-1}\leq \rho_n$$ where $\rho_1,\dots,\rho_n$ are all roots (including multiple roots which are repeated) of $P$ and $\rho'_1,\dots,\rho'_{n-1}$ are all roots of $Q$. We admit the pair $(\lambda,0)$ as interlacing for $\lambda>0$.
It is easy to show (use for example the fact that $(P,Q)$ is interlacing if and only if $Q/P$ has $n$ real poles and is decreasing on intervals without poles) that the set of all interlacing polynomials is a monoid for the product given by $$(P_1,Q_1)\cdot (P_2,Q_2)=(P_1P_2,P_1Q_2+P_2Q_1)\ .$$ The identity is of course the degenerate pair $(1,0)$.
I could find no mention of this structure in the literature on interlacing polynomials. Did it appear somewhere?
Remarks: The usual definition of interlacing pairs is slightly different: positivity of leading coefficients is not required but roots have to interlace strictly. The above definition is of course taylored for the monoid structure (which creates multiple roots by taking powers of pairs).
It is possible to work with equivalence classes of pairs by considering $(P,Q)\sim (\lambda P,\lambda Q)$ for $\lambda >0$.
The mononoid structure does not extend to "triplets" of interlacing polynomials.
