Does the following mathematical gadget have a standard name? Let $R$ be an $\mathbb{N}$-graded ring together with an $S_n$ action on each $R_n$ which are compatible in the following sense. Let $i:S_a \times S_b \rightarrow S_{a+b}$ be the standard inclusion. If $x$ is in grade $a$ with $\sigma \in S_a$ and $y$ is in grade $b$ with $\tau$ in $S_b$, then $$\sigma(a) \cdot \tau(b) = i(\sigma,\tau)(a\cdot b).$$

The main example I have in mind is the ring of G-invariant tensors.

The point is to have the right language to say that such a gadget has certain generators and relations, where I think of the symmetric group actions as unary operations.