n-ary group

In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary 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 Template:Nowrap operation is meant any map f: Gⁿ → G from the n-th Cartesian power of G to G. The axioms for an Template:Nowrap group are defined in such a way that they reduce to those of a group in the case Template:Nowrap. The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte; the first systematic account of (what were then called) 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 Template:Nowrap, 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, Template:Nowrap associativity is the equality of the n possible bracketings of a string consisting of Template:Nowrap distinct symbols with any n consecutive symbols bracketed. A set G which is closed under an associative Template:Nowrap operation is called an n-ary semigroup. A set G which is closed under any (not necessarily associative) Template:Nowrap 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 Template:Nowrap has a unique solution for x, and likewise Template:Nowrap has a unique solution. In the ternary case we generalize this to Template:Nowrap, Template:Nowrap and Template:Nowrap each having unique solutions, and the Template:Nowrap 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 Template:Nowrap semigroup which is also an Template:Nowrap quasigroup.

Structure of n-ary groups

Post gave a structure theorem for an n-ary group in terms of an associated group.Template:Rp

Identity / neutral elements

In the Template:Nowrap 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 Template:Nowrap groups for n ≥ 3 there can be zero, one, or many identity elements.

An Template:Nowrap groupoid (G, f) with Template:Nowrap, where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dörnte published the first main results: An Template:Nowrap groupoid which is reducible is an Template:Nowrap group, however for all n > 2 there exist inhabited Template:Nowrap groups which are not reducible. In some n-ary groups there exists an element e (called an Template:Nowrap 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 Template:Nowrap group containing a neutral element is reducible. Thus, an Template:Nowrap group that is not reducible does not contain such elements. There exist Template:Nowrap groups with more than one neutral element. If the set of all neutral elements of an Template:Nowrap group is non-empty it forms an Template:Nowrap subgroup.

Some authors include an identity in the definition of an Template:Nowrap group but as mentioned above such Template:Nowrap operations are just repeated binary operations. Groups with intrinsically Template:Nowrap operations do not have an identity element.

Weaker axioms

The axioms of associativity and unique solutions in the definition of an Template:Nowrap group are stronger than they need to be. Under the assumption of Template:Nowrap 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 Template:Nowrap 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.Template:Rp

Example

The following is an example of a three element ternary group, one of four such groups

$\begin{array}{ccccccccc}aaa=a& aab=b& aac=c& aba=c& abb=a& abc=b& aca=b& acb=c& acc=a\\ baa=b& bab=c& bac=a& bba=a& bbb=b& bbc=c& bca=c& bcb=a& bcc=b\\ caa=c& cab=a& cac=b& cba=b& cbb=c& cbc=a& cca=a& ccb=b& ccc=c\end{array}$

(n,m)-group

The concept of an n-ary group can be further generalized to that of an (n,m)-group, also known as a vector valued group, which is a set G with a map f: Gⁿ → G^m where n > m, subject to similar axioms as for an n-ary group except that the result of the map is a word consisting of m letters instead of a single letter. So an Template:Nowrap is an Template:Nowrap group. Template:Nowrap were introduced by G. Ĉupona in 1983.

See also

• Universal algebra

Further reading

• S. A. Rusakov: Some applications of n-ary group theory, (Russian), Belaruskaya navuka, Minsk 1998.