# iTutor Math Updates

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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?