n-ary group

The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity {{nowrap|1=(abc)de = a(bcd)e = ab(cde)}}, 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, {{nowrap|n-ary}} associativity is the equality of the n possible bracketings of a string consisting of {{nowrap|1=n + (n − 1) = 2n − 1}} distinct symbols with any n consecutive symbols bracketed. A set G which is closed under an associative {{nowrap|n-ary}} operation is called an n-ary semigroup. A set G which is closed under any (not necessarily associative) {{nowrap|n-ary}} operation is called an n-ary groupoid.

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

An n-ary group is an {{nowrap|n-ary}} semigroup which is also an {{nowrap|n-ary}} quasigroup.

Post gave a structure theorem for an n-ary group in terms of an associated group.{{rp|p=245-246}}

In the {{nowrap|2-ary}} 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 {{nowrap|n-ary}} groups for n ≥ 3 there can be zero, one, or many identity elements.

An {{nowrap|n-ary}} groupoid (G, f) with {{nowrap|1=f = (x₁ ◦ x₂ ◦ ⋯ ◦ x_n)}}, where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dörnte published the first main results: An {{nowrap|n-ary}} groupoid which is reducible is an {{nowrap|n-ary}} group, however for all n > 2 there exist inhabited {{nowrap|n-ary}} groups which are not reducible. In some n-ary groups there exists an element e (called an {{nowrap|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 {{nowrap|n-ary}} group containing a neutral element is reducible. Thus, an {{nowrap|n-ary}} group that is not reducible does not contain such elements. There exist {{nowrap|n-ary}} groups with more than one neutral element. If the set of all neutral elements of an {{nowrap|n-ary}} group is non-empty it forms an {{nowrap|n-ary}} subgroup.Wiesław A. Dudek, [https://arxiv.org/abs/0704.2749 Remarks to Głazek's results on n-ary groups], Discussiones Mathematicae. General Algebra and Applications 27 (2007), 199–233.

Some authors include an identity in the definition of an {{nowrap|n-ary}} group but as mentioned above such {{nowrap|n-ary}} operations are just repeated binary operations. Groups with intrinsically {{nowrap|n-ary}} operations do not have an identity element.Wiesław A. Dudek and Kazimierz Głazek, [https://arxiv.org/abs/math/0510185v1 Around the Hosszú-Gluskin theorem for n-ary groups], Discrete Mathematics 308 (2008), 486–4876.

The axioms of associativity and unique solutions in the definition of an {{nowrap|n-ary}} group are stronger than they need to be. Under the assumption of {{nowrap|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 {{nowrap|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.

The associativity axiom can also be given in a weaker form.{{rp|17}}

The following is an example of a three element ternary group, one of four such groups{{cite web|url=http://tamivox.org/dave/math/tern_quasi/assoc12345.html|title=Some of the fully associative ternary quasigroups|work=Ternary quasigroups|date=21 May 2019|first=Dave|last=Barber|access-date=2024-06-28}}

:$\backslash begin\{matrix\}$

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{matrix}

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 {{nowrap|(n,1)-group}} is an {{nowrap|n-ary}} group. {{nowrap|(n,m)-groups}} were introduced by G. Ĉupona in 1983.[https://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2000/1450-59320004115U.pdf On (n, m)-groups], J. Ušan, Mathematica Moravica, 2000

• Universal algebra

{{reflist}}

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

Category:Algebraic structures