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In this answer Brian M. Scott describes the following generalization of a binary associative multiplication to a ternary one: it is a function $$[\cdot,\cdot,\cdot] : G\times G \times G \to G$$ such that $$[[a,b,c],d,e] = [a, [b,d,e], [c,d,e]]$$ The reasoning is roughly that for ordinary binary associativity we have a left multiplication function $a:G \mapsto f_a : G \to G$, such that $f_a \circ f_b = f_{f_a(b)}$, while for a ternary associative operation we have a "left multiplication" $a:G \mapsto f_a: G \times G \to G$ such that $$f_a \circ (f_b, f_c) = f_{f_a (b,c)}$$

It may also probably make sense to demand different symmetrizations of this relation, like $$[a,b,[c,d,e]] = [[a,b,c],[a,b,d],e]$$ and I'm also no entirely sure if the arguments on the right are correct, so we could consider various relations of the form $$[[a,b,c],d,e] = [a, [b,d,?], [c,?,e]]$$ The reasoning here is that if we would try to define "ternary algebras", then it would make sense to have all relations given by polylinear operations (which isn't true in the example above). I believe a natural generalization to finite-dimensional ternary algebras should have relations of the form $$[[a,b,c],d,e] = \frac{1}{\mathrm{dim}}\mathrm{Tr}_w [a, [b,d,w], [c,w,e]]$$ An example of such a ternary algebra would be the space of rank 3 tensors $a_{ijk}$ with the bracket given by $$[a,b,c]_{ijk} = \sum_{pqs} a_{piq} b_{qjs} c_{skp}$$ Thus it is possible that in the case of ternary groups there should also be some averaging over all elements in place of $?$.

My question is: have such structures been studied? If so, then what are their names and relevant references? Searching on the google for "ternary operations" or "ternary algebras" doesn't yield any useful results.

EDIT: I am not asking about ternary multiplications in general, most references that I found focus on a very naive approach which is equivalent to a binary associative multiplication in most interesting cases and isn't very far in general. I am interested specifically in the associativity condition of signature $[-, -, [ -, -, -]] = [[-,-,-], [-,-,-],- ]$, which has 2 multiplications on the left, but 3 on the right. Most articles discuss signatures of the form $[-,-,[-,-,-]] = [[-,-,-],-,-]$.

One can also try to generalize the above definitions to $n$-ary algebras for arbitrary $n$, but the precise relations are even less clear.

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  • $\begingroup$ Many forms of generalized associativity have been studied. Except for hyperassociativity (which is a specialization), I am unfamiliar with the literature. If you search the Rings And Algebras section in ArXiv for combinations of ternary, generalized, and associativity, you might find a thread to pick up. If I had to ask a person for a lead, I would start with either Keith Kearnes or Shelly Wismath, as they may know someone who could answer your question. (I would be mildly surprised if they could answer it directly.) Gerhard "Hardly Doing Universal Algebra Presently" Paseman, 2017.03.29. $\endgroup$ – Gerhard Paseman Mar 29 '17 at 20:44
  • $\begingroup$ arxiv.org/abs/0812.0707 $\endgroup$ – Steve Huntsman Mar 29 '17 at 20:45
  • $\begingroup$ Relevant terms of art appear to be "associative ternary algebras" and "associative triple systems" $\endgroup$ – Steve Huntsman Mar 29 '17 at 20:46
  • $\begingroup$ It is possible that Yuri Movsisyan (might still be at Yerevan) could also give you a lead. Gerhard "If He Has Not Retired" Paseman, 2017.03.29. $\endgroup$ – Gerhard Paseman Mar 29 '17 at 20:48
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    $\begingroup$ Median algebras satisfy your ternary associativity condition: in fact, it's the third axiom in the axiomatization given here: arxiv.org/abs/1607.07747 $\endgroup$ – zeb May 28 '17 at 22:33
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A general notion of an $n$-ary associative operation (and the associated notions of $n$-ary semigroup and group) is discussed in Dudek, W. A. and Glazek, K. "Around the Hosszú–Gluskin theorem for $n$-ary groups." Disc. Math. 308, 4861 (2008). The arXiv version is https://arxiv.org/abs/math/0510185.

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  • $\begingroup$ Thank you, but their notion is entirely different from the one I describe above. They focus on the obvious generalization of associativity. $\endgroup$ – Anton Fetisov Mar 29 '17 at 21:15

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