Associativity
The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity (abc)de = a(bcd)e = ab(cde), i.e. the equality of the three possible bracketings of the string abcde in which any three consecutive symbols are bracketed. (Here it is understood that the equations hold for all choices of elements a, b, c, d, e in G.) In general, n-ary associativity is the equality of the n possible bracketings of a string consisting of n + (n − 1) = 2n − 1 distinct symbols with any n consecutive symbols bracketed. A set G which is closed under an associative n-ary operation is called an n-ary semigroup. A set G which is closed under any (not necessarily associative) n-ary operation is called an n-ary groupoid.
Identity / neutral elements
In the 2-ary case, there can be zero or one identity elements: the empty set is a 2-ary group, since the empty set is both a semigroup and a quasigroup, and every inhabited 2-ary group is a group. In n-ary groups for n ≥ 3 there can be zero, one, or many identity elements.
An n-ary groupoid (G, f) with f = (x1 ◦ x2 ◦ ⋯ ◦ xn), where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dörnte [2] published the first main results: An n-ary groupoid which is reducible is an n-ary group, however for all n > 2 there exist inhabited n-ary groups which are not reducible. In some n-ary groups there exists an element e (called an n-ary identity or neutral element) such that any string of n-elements consisting of all e's, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity e, eeae = a for every a.
An n-ary group containing a neutral element is reducible. Thus, an n-ary group that is not reducible does not contain such elements. There exist n-ary groups with more than one neutral element. If the set of all neutral elements of an n-ary group is non-empty it forms an n-ary subgroup.[4]
Some authors include an identity in the definition of an n-ary group but as mentioned above such n-ary operations are just repeated binary operations. Groups with intrinsically n-ary operations do not have an identity element.[5]