N-ary group
In mathematics, and in particular universal algebra, the concept of an n-ary group is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an operation is meant any map f: Gn → G from the n-th Cartesian power of G to G. The axioms for an group are defined in such a way that they reduce to those of a group in the case. The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte; the first systematic account of polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the Transactions of the American Mathematical Society.
Axioms
Associativity
The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity, i.e. the equality of the three possible bracketings of the string abcde in which any three consecutive symbols are bracketed. In general, associativity is the equality of the n possible bracketings of a string consisting of distinct symbols with any n consecutive symbols bracketed. A set G that is closed under an associative operation is called an n-ary semigroup. A set G'' that is closed under any operation is called an n''-ary groupoid.Inverses / unique solutions
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means has a unique solution for x, and likewise has a unique solution. In the ternary case we generalize this to, and each having unique solutions, and the case follows a similar pattern of existence of unique solutions and we get an 'n''-ary quasigroup.'''Definition of ''n''-ary group
An n-ary group is an semigroup that is also an quasigroup.Structure of ''n''-ary groups
Post gave a structure theorem for an n-ary group in terms of an associated group.Identity / neutral elements
In the 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 groups for n ≥ 3 there can be zero, one, or many identity elements.An groupoid with, where is a group is called reducible or derived from the group. In 1928 Dörnte published the first main results: An groupoid that is reducible is an group, however for all n > 2 there exist inhabited groups that are not reducible. In some n-ary groups there exists an element e such that any string of n-elements consisting of all e
An group containing a neutral element is reducible. Thus, an group that is not reducible does not contain such elements. There exist groups with more than one neutral element. If the set of all neutral elements of an group is non-empty it forms an subgroup.
Some authors include an identity in the definition of an group but as mentioned above such operations are just repeated binary operations. Groups with intrinsically operations do not have an identity element.
Weaker axioms
The axioms of associativity and unique solutions in the definition of an group are stronger than they need to be. Under the assumption of associativity it suffices to postulate the existence of the solution of equations with the unknown at the start or end of the string, or at one place other than the ends; e.g., in the case, xabcde = f and abcdex = f, or an expression like abxcde = f. Then it can be proved that the equation has a unique solution for x in any place in the string.The associativity axiom can also be given in a weaker form.