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There was a fairly recent preprint on arXiv that I caught a quick glimpse of and now can’t find again. It introduced some machinery for algebraic theories with restricted arities, perhaps introducing some operad-like notions. I remember that at least one notion concerned sets of arities $A \subseteq \mathbb{N}$ such that $a_1 + \ldots + a_k \in A$ whenever $k \in A$ and $a_1, \ldots, a_k \in A$. Can anyone recall which paper it was?
Maybe you are thinking about my talk at FMCS 2024. I talk about the set of arities on slide 4. slides. We talked quite a bit about this on Zulip. There is no preprint today because I thought that the notion wasn’t yet enough general and is probably included into an existing notion. But maybe I should try to write one if there is some interest about this notion.
Let me know if this is not what you were thinking about. If there is a preprint as you said, then it is quite intriguing to me.
The definition on your slide 4 is indeed the one that I was asking about. However, I haven’t seen your talk and I’m pretty sure I haven’t seen your slides until now either. I must have seen this notion in an arXiv paper not too long ago.
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