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As someone who has an undergraduate education in mathematics, but didn't take it any further, I have often wondered something.

Of course mathematicians like to generalize ideas. i.e. it is often better to define and write proofs for a wider scope of objects than for a specific type of object. A kind of "paradox" if you will - the more general your ideas, often the deeper the proofs (quoting a professor).

Anyway I used to often wonder about group theory especially the idea of it being a set with a list of axioms and a binary function $a\cdot b = c$. But has anybody done research on tertiary (or is that trinary/ternary) groups? As in, the same definition of a group, but with a $\cdot (a,b,c) = d$ function.

Is there such a discipline? Perhaps it reduces to standard group theory or triviality and is provably of no interest. But since many results in Finite Groups are very difficult, notably the classification of simple groups, has anybody studied a way to generalize a group in such a way that a classification theorem becomes simpler? As a trite example: Algebra was pretty tricky before the study of imaginary numbers. Or to be even more trite: the Riemann $\zeta$ function wasn't doing much before it was extended to the whole complex plane.

EDIT: Just to expand what I mean. In standard groups there is an operation $\cdot :G\times G\to G$ I am asking about the case with an operation $\cdot :G\times G\times G\to G$

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  • $\begingroup$ Could you elaborate on your definition? It reminds me of vectors: take $a,b,c\in\mathbb{R}$ and $(a,b,c)$ a vector with $a$ a scalar. Maybe $d$ comes from taking the magnitude? Are we trying to talk about a grouplike thing that operates on itself and 3-tuples of itself in a different way? $\endgroup$
    – ttt
    Dec 28, 2011 at 14:27
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    $\begingroup$ At least one person is thinking about it: web.mit.edu/kmill/www/papers/trinary_groups.pdf $\endgroup$ Dec 28, 2011 at 14:33
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    $\begingroup$ Universal algebra studies structures with $k$-ary operations. $\endgroup$
    – lhf
    Dec 28, 2011 at 14:36
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    $\begingroup$ There's a mild generalisation of group with a ternary operation (and nothing else!): it is called a heap. $\endgroup$
    – Zhen Lin
    Dec 28, 2011 at 15:05
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    $\begingroup$ Wikipedia: n-ary group. $\endgroup$ Apr 4, 2014 at 7:23

4 Answers 4

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One problem with the idea is that the most obvious generalization to ternary operations really adds nothing new:

Proposition: Let $f:G\times G\times G:\to G$, and for brevity write $[abc]$ for $f(a,b,c)$. Suppose that there is an identity element $1_G\in G$ such that $[a1_G1_G]=[1_Ga1_G]=[1_G1_Ga]$ for all $a\in G$. Suppose further that the operation is associative in the following sense: $$\big[[abc]de\big]=\big[a[bcd]e\big]=\big[ab[cde]\big]$$ for all $a,b,c,d,e\in G$. Then there is an associative binary operation $\otimes$ on $G$ such that $[abc]=a\otimes b\otimes c$ for all $a,b,c\in G$, and $1_G$ is the $\otimes$-identity.

The proof is easy. Define $\otimes:G\times G\to G$ by $a\otimes b=[ab1_G]$. Then

$$\begin{align*} (a\otimes b)\otimes c&=[(a\otimes b)c1_G]=\big[[ab1_G]c1_G\big]=\big[ab[1_Gc1_G]\big]=[abc]\\ &=\big[[abc]1_G1_G\big]=\big[a[bc1_G]1_G\big]=[a(b\otimes c)1_G]=a\otimes(b\otimes c)\;, \end{align*}$$

and $a\otimes 1_G=[a1_G1_G]=a=[1_Ga1_G]=1_G\otimes a$ for all $a,b,c\in G$. Note that this does not require any kind of generalized commutativity for the ternary operation.

I remember noticing this as an undergraduate in the late 60s. My roommate was looking at a less obvious generalization of the associative law. The generalization, as I recall, was $$\big[[abc]de\big]=\big[a[bde][cde]\big]\;.\tag{1}$$ The idea is that if $\otimes$ is a binary operation on a set $G$, one can think of each element $a\in G$ as defining a function $f_a:G\to G:x\mapsto a\otimes x$, and associativity of $\otimes$ is then the statement that $$\operatorname{Comp}(f_a;f_b)=f_{f_a(b)}\;,\tag{2}$$ where $\operatorname{Comp}$ is the composition operator. In the ternary setting think of $a\in G$ as defining a function $$f_a:G\times G\to G:\langle b,c\rangle\mapsto [abc]\;;$$ then $(1)$ becomes

$$\operatorname{Comp}(f_a;f_b,f_c)=f_{f_a(b,c)}\;,$$

generalizing $(2)$.

If I remember correctly, this approach produced more interesting structures, but I no longer remember the details.

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  • $\begingroup$ That generalization looks like it's shading towards combinators (and combinatory logic); I wonder if there's a clean translation there. $\endgroup$ Jul 3, 2015 at 5:30
  • $\begingroup$ Brian, do you have any references for the second approach? $\endgroup$ Jul 4, 2016 at 1:45
  • $\begingroup$ @Anton: I'm afraid not. That was about 50 years ago, and I don't know whether my roommate pursued the idea. It's possible that others have done something with it, but it's not my field at all, and I've no idea even where one would look. $\endgroup$ Jul 4, 2016 at 1:52
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    $\begingroup$ I guess it is worth mentioning that this answer is referred to in this MathOverflow post: Ternary associative multiplication. $\endgroup$ Sep 10, 2017 at 9:20
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    $\begingroup$ @level1807: That doesn’t actually define it: what element of $G$ is $[ab1_G]$? I see no reasonable way to define it in general, and certainly no way compatible with the name identity element: what is it leaving unchanged? One could, I suppose, experiment with defining an identity element to be an element $e$ such that $[aae]=[aea]=[eaa]=a$ for all $a\in G$, but that seems a less natural approach, and I did not look at it back then. $\endgroup$ Mar 30 at 3:20
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As Brian pointed out, the idea of ternary groups isn't that interesting, since ternary operations with identity can be shown to reduce to successively applied binary operations.

But there are some ternary group-like structures you might find interesting, and that might satisfy some of the curiosity that led you to ask this question in the first place. Someone named Dave Barber has been studying ternary quasigroups and has generated Cayley tables for a lot of them, along with names to describe their unique properties.

In constructing a ternary quasigroup, starting from a set of just 4 elements yields 55,296 possible ternary operations to choose from. Starting with 5 elements gives you 2,781,803,520 possible ternary quasigroups, and 6 elements gives 994,393,803,303,936,000 possible ternary quasigroups. Of course some of these can be can be expressed in terms of boring old binary operations, but many of them cannot! That's pretty exciting to me.

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May I suggest Ternary Mathematics Principles Truth Tables and Logical Operators 3 D Placement of Logical Elements Extensions of Boolean Algebra https://www.journalajrcos.com/index.php/AJRCOS/article/view/30166/56610

Ternary Mathematics and 3D Placement of Logical Elements Justification https://www.journalajrcos.com/index.php/AJRCOS/article/view/30257/56788

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I was also interested in this and found that n-ary groups (or grouds) are probably the nearest concept to what was asked, if not restricted to 3-ary groups.

An interesting result is that non-reducible groups have no identity elements (I suppose that means $n>2$ here):

"Groups with intrinsically n-ary operations do not have an identity element."

Wiesław A. Dudek and Kazimierz Głazek, Around the Hosszú-Gluskin theorem for n-ary groups, Discrete Mathematics 308 (2008), 486–4876.

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