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 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;W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, ''Mathematische Zeitschrift'', vol. 29 (1928), pp. 1-19. 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''.E. L. Post
Polyadic groups
''Transactions of the American Mathematical Society'' 48 (1940), 208–350.

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. (Here it is understood that the equations hold for arbitrary choices of elements ''a'',''b'',''c'',''d'',''e'' in ''G''.) 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'' which is closed under an associative operation is called an ''n''-ary semigroup. A set ''G'' which is closed under any (not necessarily associative) 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 which is also an quasigroup.

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 (''G'', ''f'') with , where (''G'', ◦) is a group is called reducible or derived from the group (''G'', ◦). In 1928 Dörnte published the first main results: An groupoid which is reducible is an group, however for all ''n'' > 2 there exist inhabited groups which are not reducible. In some ''n''-ary groups there exists an element ''e'' (called an 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 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.

Example

The following is an example of a three element ternary group, one of four such groups
:$\begin aaa = a & aab = b & aac = c & aba = c & abb = a & abc = b & aca = b & acb = c & acc = a\\ baa = b & bab = c & bac = a & bba = a & bbb = b & bbc = c & bca = c & bcb = a & bcc = b\\ caa = c & cab = a & cac = b & cba = b & cbb = c & cbc = a & cca = a & ccb = b & ccc = c \end$

(''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'': ''G''^''n'' → ''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 is an group. were introduced by G Ĉupona in 1983.On (n, m)-groups
J Ušan - Mathematica Moravica, 2000

See also

* Universal algebra

References

{{reflist

Further reading

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

Algebraic structures