Is it possible to prove associativity from $a(bc)=a(cb)$?

Question

With the given information, I'm trying to prove that an operation is both associative and commutative.

Let $(A,*)$ be a set and a binary operation such that for all $a,b,c \in A$ $$a*(b*c) = a*(c*b),$$ and for some $e \in A$ and for any $a \in A$ $$e*a = a*e = a.$$

So far, proving that $*$ is commutative has been easy, but finding a method to prove associativity has escaped me. I don't want the solution to the problem, but could anyone help in assuring me that associativity is even possible with the given assumptions.

Thanks!

Answer 1

No.

Counterexample:

$A = \{a,b,c \}$

\begin{array}{c|ccc} *&a&b&c\\ \hline a&a&b&c\\ b&b&c&c\\ c&c&c&b\\ \end{array}

$*$ is clearly commutative (and hence also $a * (b * c) = a * (c * b)$), $a$ is its identity element, but $*$ is not associative:

$b*(b*c)=b*c=c$ but $(b*b)*c=c*c=b$

Answer 2

It is not possible. Consider the operation of combining unordered binary trees, where a * b is the binary tree whose root's unordered pair of children are the roots of a and b; toss in by fiat also an identity element for this operation. This operation is clearly commutative and thus satisfies your properties, but does not satisfy associativity.

For a less abstract example, consider something like the operation $|a - b|$.