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At the end of this page: http://dogschool.tripod.com/housekeeping.html

It proves the left sided axioms using the right sided axioms. It then asks you at then end to prove the two-sided axioms using a right-sided identity axiom and a left-sided inverse axiom.

If we know only:

$e • a = a$

And

$a • a^{-1} = e$

How do we prove:

$a • e = a$

And

$a^{-1} • a = e$

Augs
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  • Identity on the right and inverse on the left does not imply the two-sided versions, as can be verified by a small example. – Tobias Kildetoft Jan 25 '16 at 19:24
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    http://math.stackexchange.com/questions/433546/is-a-semigroup-g-with-left-identity-and-right-inverses-a-group – Arnaud D. Jan 25 '16 at 19:24
  • also: http://math.stackexchange.com/questions/616577/intuition-proof-one-sided-group-definition-any-set-with-associativity-left – Chris Godsil Jan 25 '16 at 19:30

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