The late Vladimir Arnold, in 


*Arnold, V.*, [**Arithmetics of binary quadratic forms, symmetry of their continued fractions and geometry of their de Sitter world**](http://dx.doi.org/10.1007/s00574-003-0001-8), Bull. Braz. Math. Soc. (N.S.) 34, No. 1, 1-42 (2003). [ZBL1044.11016](https://zbmath.org/?q=an:1044.11016).


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 property** reasonable, or would it be confusing with something else ?
> 
> 
> 


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



Name of a group-like structure
$n$-group property is not a good idea, because it's a substructure (and also because groups have inverses). One option is $n$-subsemigroup (or any obvious variant such as $n$-fold subsemigroup, $[n]$-subsemigroup, if any reason to do so)...


As mentioned in the comments, an $n$-subsemigroup with unit is just the same as a submonoid, so no need to define $n$-submonoid.


[Note that it's the name of a **sub**structure, not of a structure. Finding axioms for the structure itself is not obvious, and possibly not clearly defined (one way to define it, using the language of universal algeba, would be to characterize, in sets endowed with an $n$-ary law, the subvariety generated by semigroups with their $n$-ary law). For instance, there is a classical notion of Lie triple system.]


