Talk:Inverse in Monoid is Unique

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The paragraph from "Similarly:" seems unnecessary... -- TheSpleen (talk) 16:03, 8 September 2013 (UTC)

There is a reason, I just can't think what it is at the moment. --prime mover (talk) 16:24, 8 September 2013 (UTC)
It's just that if you read the second equation system backwards and switch $y$ and $z$, you get exeactly the same. -- TheSpleen (talk) 17:09, 8 September 2013 (UTC)
Yes all right it's not necessary, but it's informative for a beginner. --prime mover (talk) 06:07, 9 September 2013 (UTC)
I think the original rationale for including both was because $\circ$ isn't commutative (under the monoid axioms); however, I think that since $\circ$ commutes when one argument is the identity element (by the definition of the identity element, $x\circ 1 = 1 \circ x = x$), this clarification isn't necessary. Luolimao (talk) 05:45, 12 February 2014 (UTC)


I put the mergeto the corollary because this theorem is about magmas that are associative and have an identity element. The corollary is about monoids. They are exactly the same: a monoid is an associative magma with identity. Hence the merge. --barto (talk) (contribs) 15:24, 4 November 2017 (EDT)

Good call, my bad. — Lord_Farin (talk) 16:46, 4 November 2017 (EDT)
This page was deliberately written so as to refer to a general algebraic structure which may or may not be closed. A monoid is specifically closed. Hence the explicit difference between one and the other.
So can we go back to how this page was originally?
No, let me rephrase that, we are going back to how this page was originally. --prime mover (talk) 17:58, 4 November 2017 (EDT)
Oh right, thanks. --barto (talk) (contribs) 03:42, 5 November 2017 (EST)

A subtle distinction worth preserving indeed. I already wondered how it got to be set up like this. — Lord_Farin (talk) 11:42, 5 November 2017 (EST)

Wait... In order for $\circ$ to be associative, $S$ has to be closed: $(x\circ y)\circ z$ only makes sense if $x\circ y\in S$. See Definition:Associative/Algebraic Structure. --barto (talk) (contribs) 12:12, 5 November 2017 (EST)

I'll move the first line of the proof to the theorem statement, because the theorem really doesn't make any sense without requiring that $S$ has identity. --barto (talk) (contribs) 12:09, 5 November 2017 (EST)

No, please leave it as it is. There exists a definition of inverse which does not depend on there being an identity. I want to preserve that. --prime mover (talk) 12:14, 5 November 2017 (EST)
Ehm okay.. I'm curious which definition, something not yet at ProofWiki? --barto (talk) (contribs) 12:18, 5 November 2017 (EST)
Definition:Inverse Semigroup -- I think I got it off Twikipedia, have no references to it in any of my own hardcopies. --prime mover (talk)
Well, in an inverse semigroup the inverse is unique by definition/assumption. It is unrelated. --barto (talk) (contribs) 12:40, 5 November 2017 (EST)
This whole category of mathematics was set up by someone who lied about his capabilities. Delete the lot and start all over again, it's complete shit. --prime mover (talk) 12:25, 5 November 2017 (EST)

Merge done. (associative assumes closed) --barto (talk) (contribs) 15:24, 5 November 2017 (EST)

It appears that there are other pages where the same distinction (which it isn't) has been made. I see that they're all based on Warner. He actually does only consider closed structures at that point (and indeed defines associativity only for closed structures, as it should be). --barto (talk) (contribs) 15:46, 5 November 2017 (EST)