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.



Graded rings with compatible S_n actions
Steven Sam and Andrew Snowden and their other collaborators call these `twisted commutative algebras' and have been having fun writing papers about properties of generators and similar. 


But predating this, any topologist of a certain sort would call this a monoid in the category of symmetric abelian groups. If you do the same construction in simplicial sets (or topological spaces), an example is the sphere spectrum, with nth space $S^n$. Modules over this object are what are called symmetric spectra, and serve as one of the main models of modern day stable homotopy theory. (This was an observation by Jeff Smith in the mid 1990s.)


