nLab n-ary group

Idea

A generalization of associative quasigroup to n-ary operations.

Definition

Given a natural number $n$, an $n$-ary group is a set $S$ that both an n-ary semigroup and an n-ary quasigroup.

Remark

A binary group is not a group. Binary groups are associative quasigroups, as they can be empty.

Properties

Definition

An n-ary identity element is an element $e$ such that any string of $n$ elements consisting of all $e$‘s, apart from one place, is mapped to the element at that place.

Theorem

Every n-ary group with an n-ary identity element is a group, with the n-ary group operation being simply repeated application of a group’s binary operation.

Remark

The above theorem is why most authors typically do not require the existence of $n$-ary identity elements in the definition of $n$-ary groups, and why binary groups are just associative quasigroups.

See also

• associative quasigroup

• n-ary magma

• n-ary semigroup

• n-ary quasigroup

• group

References

• W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Mathematische Zeitschrift, vol. 29 (1928), pp. 1-19.

• E. L. Post, Polyadic groups, Transactions of the American Mathematical Society 48 (1940), 208–350.

• W. A. Dudek, “On some old and new problems in n-ary groups”, Quasigroups and Related Systems, 8 (2001), 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.

• Wikipedia, n-ary group