Every commutative monoid can be extended to a group. Called the Grothendieck group completion.
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It's a great thing! But don't let the word "extended" fool you: the homomorphism from the monoid to the group doesn't need to be one-to-one!
I totally let the word "extended" fool me. I've been bamboozled yet again.
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I would not use the word "extended" in this situation. The map from a commutative monoid to its Grothendieck group is one-to-one iff the monoid is "cancellative", meaning x+y=x'+y implies x=x'.
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E.g. if your monoid has idempotents, there's no way these can be mapped faithfully into the group completion, so you lose injectivity. For a similar reason I asked this question earlier, to see just how far from capacity cancellativity a monoid can get:https://twitter.com/silvascientist/status/1141155911529984000?s=19 …
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