I am not as familiar with operad terminology as I'd like to be, so I might be missing some well-known term in the area. If so, I'd appreciate any pointers to the correct terms.

Consider the following two objects:

- The free associative operad with one binary operation over $\mathbb{Q}$. What I mean is that we are working with $\mathbb{Q}$-vector spaces, and consider the operad where the generating set is an operation of arity 2 and the relation is the associativity relation.
- The free associative operad with one binary operation over $\mathbb{Z}$. What I mean is that we are working with $\mathbb{Z}$-modules (aka abelian groups), and consider the operad where the generating set is an operation of arity 2 and the relation is the associativity relation.

(2) can be viewed as a $\mathbb{Z}$-suboperad of (1). I am interested in defining an intermediate $\mathbb{Z}$-suboperad of (1). The goal is to allow for a limited amount of division based on the degree.

The intermediate suboperad I want to define is the suboperad of (1) generated by the following infinite set of operations: for each prime number $p$, consider:

$$(x_1,x_2,\dots,x_p) \mapsto \frac{1}{p}(x_1x_2 \dots x_p)$$

The representations of the operad (1) correspond to associative $\mathbb{Q}$ algebras and the representations of the operad (2) correspond to associative $\mathbb{Z}$-algebras (also known as associative rings). The representations of the intermediate operad defined above are associative rings with additional "multiply them all and divide by $p$" operations associated to strings of length $p$ for every prime $p$.

The intermediate object I have defined above is closely related to the nature of denominators that appear in the Baker-Campbell-Hausdorff formula. If we perform a similar construction for the Lie operad instead of the associativity operad (UPDATE: There is a slight issue, in that the assumption of the *alternating* condition rather than *skew symmetry* is a non-operadic identity in so far as it involves a repeated variable, and this could be an issue if we are working over a base where 2 is non-invertible), then the representations of that operad will be the "Lie algebras" in a suitable generalization of the Lie correspondence or Lazard correspondence (there are some messy details I am not getting into here).

[As a quick illustration, note that the Baker-Campbell-Hausdorff formula has a denominator of 12 for its degree three terms. To make sense of a denominator of 12, note that we can separately make sense of denominators of 3 and 4 for length three Lie products: the denominator of 3 makes sense directly setting $p = 3$, and the denominator of 4 makes sense by consider $\frac{1}{2}[x,\frac{1}{2}[y,z]]$. We can combine these using the Euclidean algorithm to get a denominator of $1/12$ (in this case, $1/12 = 1/3 - 1/4$). There are general bounds on the denominators that appear in the Baker-Campbell-Hausdorff formula that guarantee that the denominators can always be achieved.]

Additionally, the general theme behind this sort of operad seems well-suited to situations where we want to introduce divisibility by certain primes but only in high degrees dependent on the prime. This might make it suitable to the study of things like the representation theory of symmetric groups, though I might be barking up the wrong tree here.

I'd like to know if operads of this sort have been studied before, and if so, I'd like any pointers to the appropriate terms and references. If not, I would appreciate any views on whether this is useful as an intermediate framework between the extremes of $\mathbb{Z}$-operads and $\mathbb{Q}$-operads.