N-ary group

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In mathematics, an n-ary group (also n-group, polyadic group or multiary group) is a generalization of a group to a set G with a n-ary operation instead of a binary operation.^[1] The axioms for an n-ary group are defined in such a way as to reduce to those of a group in the case n = 2.

Axioms

Associativity

The easiest axiom to generalize is the associative law. Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. n-ary associativity is a string of length n+(n-1) with any n adjacent elements bracketed. A set G with a closed n-ary operation is an n-ary groupoid. If the operation is associative then it is an n-ary semigroup.

Inverses / Unique Solutions

The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means ax = b has a unique solution for x, and likewise xa = b has a unique solution. In the ternary case we generalize this to abx = c, axb = c and xab = c each having unique solutions, and the n-ary 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 n-ary semigroup which is also an n-ary quasigroup.

Identity / Neutral elements

In the 2-ary case, i.e. for an ordinary group, the existence of an identity element is a consequence of the associativity and inverse axioms, however in n-ary groups for n ≥ 3 there can be zero, one, or many identity elements.

An n-ary groupoid (G, ƒ) with ƒ = (x₁ ◦ x₂ ◦ . . . ◦ x_n), where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dornte published the first main results: An n-ary groupoid which is reducible is an n-ary group, however for all n > 2 there exist 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.^[2]

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.^[3]

Weaker axioms

The axioms of associativity and unique solutions in the definition of an n-ary group are stronger than they need to be. Under the assumption of n-ary 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 6-ary 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.^[4] The associativity axiom can also be given in a weaker form - see page 17 of "On some old and new problems in n-ary groups".^[1]

Example

The following is an example of a three element ternary group, one of four such groups^[5]

$aaa=a,aab=b,aac=c,aba=c,abb=a,abc=b,aca=b,acb=c,acc=a,$

$baa=b,bab=c,bac=a,bba=a,bbb=b,bbc=c,bca=c,bcb=a,bcc=b,$

$caa=c,cab=a,cac=b,cba=b,cbb=c,cbc=a,cca=a,ccb=b,ccc=c.$

References

↑ 1.0 1.1 Dudek, W.A. (2001), "On some old and new problems in n-ary groups", Quasigroups and Related Systems 8: 15–36 .
↑ Wiesław A. Dudek, Remarks to Głazek's results on n-ary groups, Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.
↑ Wiesław A. Dudek and Kazimierz Głazek, Around the Hosszú-Gluskin theorem for n-ary groups, Discrete Mathematics 308 (2008), 486–4876.
↑ E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.
↑ http://home.comcast.net/~tamivox/dave/math/tern_quasi/assoc1234.html

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

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