The late Vladimir Arnold, in

*Arnold, V.*, **Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world**, Bull. Braz. Math. Soc. (N.S.) 34, No. 1, 1-42 (2003). ZBL1044.11016.

introduced (in the context of binary quadratic forms, but the concept is general) the following definition (in multiplicative notation): a subset $\mathcal{A}$ of a semigroup $\mathcal{S}$ has the **trigroup property** if for any triplet $(a_1,a_2,a_3)\in\mathcal{A}^3$ one always has $a_1a_2a_3\in\mathcal{A}$.
Remark: of course $a_1a_2$, $a_1a_3$ or $a_2a_3$ might not be in $\mathcal{A}$ (that's the point).

I have been working on another topic where the following generalisation pops up naturally: a subset $\mathcal{A}$ of a semigroup $\mathcal{S}$ has the **$n$-group property** if for any $n$-tuple $(a_1,\dots ,a_n)\in\mathcal{A}^n$ one always has $\prod_{i=1}^na_i\in\mathcal{A}$.
Remark: some of the smaller products $\prod_{j\in J,1<|J|<n}a_j$ might not be in $\mathcal{A}$.

Question: (a) has that generalisation been already studied, perhaps with a different name ? As far as I can tell from papers citing Arnold's it's not the case, but maybe it came up before his paper. (b) in the event it has no name yet, is the name$n$-group propertyreasonable, or would it be confusing with something else ?

[*Edited several times to take into account the comments.*]