---
tag: 7ESVWM5KM3
kind: statement
role: proposition
name: Identity / neutral elements
sources:
  - wikipedia: en:23340612@r1359521036
    locator: N-ary group, section Axioms > Identity / neutral elements
    note: adopted
    mention: wikipedia://en/23340612#5WCX
    sections: [Statement]
inventory: [VF1W49P9KE]
---

## Statement

In the 2-ary case, there can be zero or one identity elements: the [[empty set|0ECVK59Y03]] 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* = (*x*<sub>1</sub> ◦ *x*<sub>2</sub> ◦ ⋯ ◦ *x*<sub>*n*</sub>), where (*G*, ◦) is a group is called reducible or derived from the group (*G*, ◦). In 1928 Dörnte  published the first main results: An *n*-ary groupoid that is reducible is an *n*-ary group, however for all *n* > 2 there exist inhabited *n*-ary groups that 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.

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|07WYQPB8E3]].