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In mathematics , and in particular universal algebra , the concept of n -ary group also called n -group or multiary group is a generalization of the concept of group to a set G with an n -ary operation instead of a binary operation.

The easiest axiom to generalize is the associative law.

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

elementary set theory - Are all n-ary operators simply compositions of binary operators? - Mathematics Stack Exchange

The inverse axiom is generalized as follows: An n -ary group is an n -ary semigroup which is also an n -ary quasigroup.

In the 2-ary case, i.

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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.

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.

Binary operation - Wikipedia

Groups with intrinsically n -ary operations do not have an identity element. 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. Then it can be proved that the equation has a unique solution for x in any place in the string. The following is an example of a three element ternary group, one of four such groups [6].

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