Reducible multiary group

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

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