Talk:N-ary group

Search terms
Best google scholar search term seems to be:

"ternary groups" OR "n-ary groups" OR "polyadic groups"

Adding "n-groups" just adds too many irrelevant hits. Charvest (talk) 19:20, 24 June 2009 (UTC)

Example
Is the following based on the set {a,b,c} a ternary group? Does it have any other name? It seems to satisfy the associative requirement. --Rumping (talk) 15:46, 8 July 2009 (UTC)


 * 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.

I have made some changes to the article to highlight the definition of n-ary group. It must satisfy both associativity and each equation in one unknown must have a unique solution. (No need of an identity or inverse if n>2). I haven't checked whether the above example satisfies both these conditions but ternary associativity means all strings of length 5 must be checked which is 3^5=243 possibilities, although the "old and new" paper describes a weaker form of associativity which is sufficient. Charvest (talk) 18:04, 8 July 2009 (UTC)

I verified the example above to be a ternary group. If you need more examples, feel free to use mine: http://home.comcast.net/~tamivox/dave/math/tern_quasi/assoc1234.html discussed at http://home.comcast.net/~tamivox/dave/math/tern_quasi/index.html Lemonroe (talk) 13:56, 20 April 2010 (UTC)

Identity element
I removed remarks about an identity element from the definition. They belong elsewhere; I put the essence into the section on identity/neutral elements. (I also thought they were wrong, but I was thinking of n-quasigroups, so I was mistaken.) Zaslav (talk) 05:22, 12 June 2010 (UTC)