Comments by "MC116" (@angelmendez-rivera351) on "Proof that the identity element of a group is unique" video.

  1. In fact, with this proof, you can prove a much stronger statement. If (M, •) is a magma with left-identity L and right-identity R, then L = L•R = R, and so L = R = e is the unique two-sided identity element in M, and (M, •) is called a unital magma.
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  2.  @randomblueguy  Every group has the cancellation property, but not every magma with an identity element that has the cancellation property is a group. Also, not every magma with an identity element has the cancellation property.
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  3.  @VinayKumar-nf6sd  See what I said above.
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  4. If a magma (M, •) has a left identity L and right identity R, then L = R, because L = L•R = R. As such, L = R = e is a two-sided identity element. The two-sided element is unique by consequence of this very same proof. If e is a two-sided identity element and f is a two-sided identity element, then e = e•f = f. That being said, a magma can have multiple left-identity elements or multiple right-identity elements if no other-sided identity elements exist. However, if the magma has the cancellative property, which follows if the magma is a quasigroup, then the one-sided identity elements are unique. If you have associativity, cancellativity, and unique one-sided identity elements, then you have a unique two-sided identity element, and thus this is a monoid.
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