symmetric monoidal (∞,1)-category of spectra
A generalization of associative quasigroup to n-ary operations.
Given a natural number $n$, an $n$-ary group is a set $S$ that both an n-ary semigroup and an n-ary quasigroup.
A binary group is not a group. Binary groups are associative quasigroups, as they can be empty.
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.
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.
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.
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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
Created on August 4, 2022 at 04:48:08. See the history of this page for a list of all contributions to it.