N-ary group
Encyclopedia
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. The axioms for an n-ary group are defined in such a way as to reduce to those of a group in the case .
An n-ary groupoid (G, ƒ) with ƒ = (x1 ◦ x2 ◦ . . . ◦ xn), 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an n-ary group (also n-group, polyadic group or multiary group) is a generalization of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
to a set G with a n-ary operation
Arity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
instead of a binary operation. The axioms for an n-ary group are defined in such a way as to reduce to those of a group in the case .
Associativity
The easiest axiom to generalize is the associative law. Ternary associativity is , 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 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 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 ◦ . . . ◦ xn), 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
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. Groups with intrinsically n-ary operations do not have an identity element.