Natural associative law for a ternary "group"? Suppose one were to define a group-like structure based on a set $G$
with a ternary (rather than binary) operator $g( a, b, c ) = \left< a, b, c \right>$.
One possible definition for the associative law is that
$$
\left< \left< a, b, c \right>, d, e \right>
=
\left< a,\left< b, c, d \right>, e \right>
=
\left< a, b, \left< c, d, e \right> \right>
\;.
$$
Is there some algebraic structure defined along these lines that
has been investigated?
If so, I'd appreciate a name & pointer.
Are there other natural definitions of ternary associativity? 
And definitions of ternary identity laws?
(Not entirely relevant, but I came upon this thinking about medians and centroids,
and I am just looking for possible models...)
 A: There are many operads that are generated in degree three  (the operations) and have relations in degree 5, like the ones you are thinking. One that has been studied in pretty detail is the notion of Ternary Jordan Algebra. Over Vect this is a vector space $A$ with a ternary operation $\langle a,b,c\rangle$ which satisfies a symmetry condition $\langle a,b,c \rangle = \langle c, a, b\rangle$  and a degree five relation 
$\langle a, b, \langle c,d,e \rangle\rangle + \langle c, \langle b, a, d \rangle, e \rangle = \langle \langle a,b,c \rangle, d, e \rangle + \langle c, d, \langle a,b,e\rangle \rangle $
This plays a role in the study of Jordan Algebras and symmetric spaces. You can take a look at Neher's paper [1].
There are versions for higher Jacobi as well. The book of Loday and Vallette [2] will also give you many examples of algebraic structures with ternary operations and different versions of the higher "associativity" condition.   
[1] Neher, on the classification of Lie and Jordan triples Commun. Alg. 13 (1985) 2615--2667
[2] Jean-Louis Loday, Bruno Vallette "Algebraic Operads"
A: That is the same definition of ternary associativity that I have seen. Such ternary operations are discussed in the papers [1],[2]. Furthermore, the paper [3] surveys some results on n-ary groups.
Ternary associativity can be used to create binary associative operations. This is illustrated in the following proposition. Take note that all the listed identities in the following proposition are some forms of associativity.

$\mathbf{Proposition}$ Suppose that $t$ is a ternary operation on a
  set $X$. Then define operations $*,\#$ on $X^{2}$ by letting
  $(a,b)*(c,d)=(t(a,b,c),d)$ and $(a,b)\#(c,d)=(a,t(b,c,d))$. Then 
  
  
*
  
*$(X^{2},*,\#)$ always satisfies the identity
  
  
  $$((a,b)*(c,d))\#(e,f)=(a,b)*((c,d)\#(e,f)).$$

  
*The following are equivalent.
  
  
  i. $t$ satisfies the identity  $$t(t(a,b,c),d,e)=t(a,b,t(c,d,e)).$$
ii. $*$ is associative.
iii. $\#$ is associative.

  
*The following are equivalent.
  
  
  i. $t$ satisfies the identity $t(t(a,b,c),d,e)=t(a,t(b,c,d),e)$.
ii. $(X^{2},*,\#)$ satisfies the identity
  $$((a,b)\#(c,d))*(e,f)=((a,b)*(c,d))*(e,f).$$
iii. $(X^{2},*,\#)$ satisfies the identity
$$(a,b)\#((c,d)*(e,f))=((a,b)\#(c,d))\#(e,f).$$

  
*The following are equivalent.
  
  
  i. $t$ satisfies the identity $t(a,t(b,c,d),e)=t(a,b,t(c,d,e))$
ii. $(X^{2},*,\#)$ satisfies the identity
$$(a,b)*((c,d)*(e,f))=((a,b)\#(c,d))*(e,f).$$
iii. $(X^{2},*,\#)$ satisfies the identity
$$(a,b)\#((c,d)*(e,f))=(a,b)\#((c,d)\#(e,f)).$$

I proved the above proposition just now to answer this question, so I am unsure if reducing ternary associative operations to binary associative operations has been investigated before (let me know if you find any reference where reducing ternary associative operations  to binary associative operations has been studied before). It seems like the above proposition generalizes to n-ary operations as well (I have not verified this yet). Unfortunately, the above proposition does not seem to work very well for having two sided identities for either of the operations $*$ or $\#$. The following proposition (with a trivial proof) relates ternary identities to binary identities.

$\mathbf{Proposition}$ Let $t$ be a ternary operation on a set $X$,
  and define binary operations $*,\#$ on $X^{2}$ as before by
  letting $(a,b)*(c,d)=(t(a,b,c),d),(a,b)\#(c,d)=(a,t(b,c,d))$. Let
  $e_{1},e_{2}\in X$. Then
  
  
*
  
*$(e_{1},e_{2})*(a,b)=(a,b)$ for each $a,b\in X$ if and only if $t(e_{1},e_{2},a)=a$.
  
*$(a,b)\#(e_{1},e_{2})=(a,b)$ if and only if $t(b,e_{1},e_{2})=b$.

[1] Ataguema, H.; Makhlouf, A., Deformations of ternary algebras, Journal of Generalized Lie Theory and Applications, vol. 1, (2007), 41–45.
[2] Ataguema, H.; Makhlouf, A., Notes on cohomologies of ternary algebras of associative type, Journal of Generalized Lie Theory and Applications, vol. 3, no. 3 (2009) 154–174.
[3] Remarks to Glazek's results on n-ary groups
http://arxiv.org/pdf/0704.2749v1.pdf
