Reducible multiary group

Definition

Recall that a multiary group is a $n$-ary group for some $n\geq 2$. A $n$-ary group with multiplication $f:G^{n}\to G$ is termed reducible if it satisfies the following equivalent conditions:

No.	Shorthand	Statement with symbols	Note
1	arises from a group	there exists a group structure on $G$ such that $f(a_{1},a_{2},\dots ,a_{n})=a_{1}a_{2}\dots a_{n}$ for all (possibly repeated) $a_{1},a_{2},{\dot {,}}a_{n}\in G$, with the multiplication on the right being as per the group structure. For more on this, see group is n-ary group for all n	If we're assuming this, we don't need to check the $n$-ary associativity and unique solution conditions; they follow automatically.
2	existence of neutral element	there exists a neutral element $e\in G$ for $f$: $f$ evaluated at any tuple where $n-1$ of the entries are equal to $e$ and the remaining entry is $a\in G$, gives output $a$. This is true regardless of where we place $a$ and is also true if $a=e$.	Note that we do need to separately check the associativity and unique solution conditions.

Equivalence of definitions

Further information: equivalence of definitions of reducible multiary group

Facts

• Characterization of subgroup of neutral elements of reducible multiary group
• Equivalence of definitions of reducible multiary group
• Groups giving same reducible multiary group are isomorphic