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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...)

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    $\begingroup$ Loday discusses «totally associative algebras» (with an $n$-ary operation) which satisfy exactly those relations in his book about algebraic operads. $\endgroup$ – Mariano Suárez-Álvarez Nov 5 '14 at 0:45
  • $\begingroup$ @MarianoSuárez-Alvarez: Thanks for the term "totally associative algebra." And Reimundo Heluani cited the same book. $\endgroup$ – Joseph O'Rourke Nov 5 '14 at 1:08
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    $\begingroup$ I added a universal algebra tag since this question deals with very general algebraic structures of arity higher than two. $\endgroup$ – Joseph Van Name Nov 5 '14 at 1:32
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What you need is the keyword "polyadic groups". A polyadic group is a non-empty set $G$ equipped with an associative $n$-ary operation $f:G^n\to G$ such that for all $a_1, \ldots, a_{n}$ and $b\in G$, the equations $$f(a_1, \ldots, a_{i-1}, x, a_{i+1}, \ldots, a_n)=b, (1\leq i\leq n)$$ have (unique) solution for $x$. The simplest examples are polyadic groups derived from ordinary groups: Let $(G, \cdot)$ be a group and define an $n$-ary operation on $G$ by $$f(x_1, \ldots, x_n)=x_1x_2\cdots x_n.$$ Then $(G,f)$ becomes a polyadic group which is called the polyadic group derived from $(G, \cdot)$ and it is denoted by $\mathrm{der}(G, \cdot)$. It is easy to see that only polyadic groups of this type have identity element. In general, the structure of a polyadic group can be described in terms of ordinary groups and their automorphisms. If $(G, f)$ is a polyadic group, then there exists a binary operation $\ast$ on $G$ such that $(G, \ast)$ is a group, and there is an automorphism $\theta\in \mathrm{Aut}(G, \ast)$ and an element $b\in G$ such that $\theta^{n-1}(x)=bxb^{-1}$ and $\theta(b)=b$ and $$f(x_1, \ldots, x_n)=x_1\ast\theta(x_2)\ast\cdots \ast\theta^{n-1}(x_n)\ast b$$ and the converse is also true. So, the polyadic group is denoted in general by $\mathrm{der}_{b, \theta}(G, \ast)$.

Many problems concerning polyadic groups are solved until now; Emil Post proved that any polyadic group $(G, f)$ has a covering group $G^{\ast}$ which is generated by $G$ and inside $G^{\ast}$ we have $$f(x_1, \ldots, x_n)=x_1x_2\ldots x_n$$. Moreover $G^{\ast}$ contains the retract group of $(G, f)$ as a normal subgroup and the corresponding quotient is cyclic of order $n-1$. Here is a list of some interesting works done for polyadic groups and you can find appropriate references using keywords in the internet.

1- The variety of polyadic groups and its subvarieties is studied by Dudek, Artamonov, and ... .

2- The universal Post cover and retracts are studied by Michalisky and Dudek.

3- Free polyadic groups are studied by Artamonov.

4- Representation theory of polyadic groups are studied by me and Dudek.

5- The structure of homomorphisms, automorphisms, and an isomorphism problem for polyadic groups, studied by me and Khodabandeh.

6- Simple polyadic groups are characterized by me and Khodabandeh.

Similar structures are also studied during the past decades, among them are $n$-ary generalization of Lie algebras (Fillipov algebras) and $(m,n)$-rings. Here is a list of some useful references:\

  1. Dudek W. A. Remarks on n-groups // Demonstr. Math. 1980. V. 13. P. 165–181.

  2. Post E. L. Polyadic groups // Trans. Amer. Math. Soc. 1940. V. 48. P. 208–350.

  3. Dornte W. Unterschungen ¨uber einen verallgemeinerten Gruppenbegriff // Math. Z. 1929. Bd 29. S. 1–19.

  4. Kasner E. An extension of the group concept // Bull. Amer. Math. Soc. 1904. V. 10. P. 290–291.

  5. Galmak A. M. N-ary groups. Gomel: Gomel Univ. Press, 2003.

  6. Dudek W. A. Varieties of polyadic groups // Filomat. 1995. V. 9. P. 657–674.

  7. Galmak A. M. Remarks on polyadic groups // Quasigroups Relat. Syst. 2000. V. 7. P. 67–70.

  8. Gleichgewicht B., Glazek K. Remarks on n-groups as abstract algebras // Colloq. Math. 1967. V. 17. P. 209–219.

  9. Dudek W. A., Michalski J. On retract of polyadic groups // Demonstr. Math. 1984. V. 17. P. 281–301.

  10. Dudek W. A., Glazek K. Around the Hossz´u–Gluskin theorem for n-ary groups // Discrete Math. 2008. V. 308. P. 4861–4876.

  11. Dudek W. A., Michalski J. On a generalization of Hossz´u theorem // Demonstr. Math. 1982. V. 15. P. 437–441.

  12. Hosszu M. On the explicit form of n-groups // Publ. Math. 1963. V. 10. P. 88–92.

  13. Ushan J. Congruences of n-group and of associated Hossz´u–Gluskin algebras // Novi Sad. J. Math. 1998. V. 28. P. 91–108.

  14. Dudek W. A., Shahryari M. Representation theory of polyadic groups // Algebr. Represent. Theory. 2012. V. 15. P. 29–51.

  15. Shahryari M. Representations of finite polyadic groups // Commun. Algebra. 2012. V. 40. P. 1625–1631.

  16. Khodabandeh H., Shahryari M. On the representations and automorphisms of polyadic groups // Commun. Algebra. 2012. V. 40. P. 2199–2212.

  17. Khodabandeh H., Shahryari M. Simple polyadic groups// Siberian Math. Journal, 2014.

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  • $\begingroup$ Could you please state explicitly what “associative $n$-ary operation” means here? $\endgroup$ – Emil Jeřábek Nov 6 '14 at 14:00
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    $\begingroup$ Per en.wikipedia.org/wiki/N-ary_group Ternary associativity is (abc)de = a(bcd)e = ab(cde), i.e. the string abcde with any three adjacent elements bracketed. n-ary associativity is a string of length (n)+(n-1) with any n adjacent elements bracketed. $\endgroup$ – Snor Nov 6 '14 at 15:43
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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"

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  • $\begingroup$ Thanks for these references. (Incidentally, Algebraic Operads is US$149 @Springer!) $\endgroup$ – Joseph O'Rourke Nov 5 '14 at 1:07
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    $\begingroup$ Oh my! that's steep. There used to be a first draft in Vallette's homepage. Try this link math.unice.fr/~brunov/Operades.html $\endgroup$ – Reimundo Heluani Nov 5 '14 at 1:15
  • $\begingroup$ Wonderful, Reimundo---Draft still there! $\endgroup$ – Joseph O'Rourke Nov 5 '14 at 1:30
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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

  1. $(X^{2},*,\#)$ always satisfies the identity

$$((a,b)*(c,d))\#(e,f)=(a,b)*((c,d)\#(e,f)).$$

  1. 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.

  1. 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

$$(a,b)\#((c,d)*(e,f))=((a,b)\#(c,d))\#(e,f).$$

  1. 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

  1. $(e_{1},e_{2})*(a,b)=(a,b)$ for each $a,b\in X$ if and only if $t(e_{1},e_{2},a)=a$.

  2. $(a,b)\#(e_{1},e_{2})=(a,b)$ 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

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  • $\begingroup$ Quote from [3]: "ternary algebraic operations were introduced already in the XIXth century by A. Cayley"! $\endgroup$ – Joseph O'Rourke Nov 5 '14 at 1:37

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