The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte;[2] the first systematic account of (what were then called) polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the Transactions of the American Mathematical Society.
(Here it is understood that the equations hold for all choices of elements a, b, c, d, e in G.) In general, n-ary associativity is the equality of the n possible bracketings of a string consisting of n + (n − 1) = 2n − 1 distinct symbols with any n consecutive symbols bracketed.
The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means ax = b has a unique solution for x, and likewise xa = b has a unique solution.
In the ternary case we generalize this to abx = c, axb = c and xab = c each having unique solutions, and the n-ary case follows a similar pattern of existence of unique solutions and we get an n-ary quasigroup.
[5] The axioms of associativity and unique solutions in the definition of an n-ary group are stronger than they need to be.
Under the assumption of n-ary associativity it suffices to postulate the existence of the solution of equations with the unknown at the start or end of the string, or at one place other than the ends; e.g., in the 6-ary case, xabcde = f and abcdex = f, or an expression like abxcde = f. Then it can be proved that the equation has a unique solution for x in any place in the string.