Can anybody do better than this? Say I define a specific type of algebraic structure (I guess, a commutative monoid) with the following axioms: 1) There exists an identity element in the structure (G) e such that for all a in G, ae = a. 2) For all a, b, and c in G, (ab)c = b(ac) And from only these we can deduce the commutative and associative axioms! ab = (ab) = (ab)e (identity axiom) = b(ae) (axiom #2) = b(a) (identity axiom) = ba And thus ab = ba. And now for the associative property, (ab)c = (ba)c (commutative property) = a(bc) (axiom #2) Is there any statement that would deduce both the commutative and associative properties without the need of an identity element?