Talk:Grothendieck group

Correct title?

Is it really correct to call the simple construction (of the universal covering group of a monoid) the Grothendieck group? Every time I've seen it presented, it was done so without the Grothendieck name attached to it, whereas the Grothendiek group is only discussed for Abelian categories, and not just on plain monoids (which is more simply the "universal abelian group" or something like that. linas 19:36, 1 April 2007 (UTC)[reply]

This came up in discussion on Talk:Semigroup. I would like to expand on the definition of the simpler thing, which I'm afraid might make the article very long, kind of drowning out the bit about the abelian category. Thus, perhaps, would it be agreeable to split this article into two, assuming that a name can be found for the "simpler thing"? (I don't know what it should be called, and notes that I have do not give it a name). linas 19:42, 1 April 2007 (UTC)[reply]

All of which is contradicted by the planetmath article, which defines only the "simpler" thing. Oh well. linas 13:07, 2 April 2007 (UTC)[reply]

The "simpler thing" is known as "group completion (of a monoid)" DesolateReality (talk) 12:24, 26 March 2009 (UTC)[reply]

It's an old comment, but I agree with linas. I think the simpler construction is due to Ore, well before Grothendieck, and is only attributed to Grothendieck by people who otherwise don't study semigroups. -- Walt Pohl (talk) 19:54, 16 January 2011 (UTC)[reply]

Cancellation

It seems pretty obvious that cancellation is necessary for this construction. So why is there no mention of it? Thehotelambush (talk) 22:43, 30 September 2008 (UTC)[reply]

Never mind, the +k accounts for this. Thehotelambush (talk) 04:27, 9 May 2009 (UTC)[reply]

Cancellativity is a necessary condition for a semigroup to be embedded in a group (see Wikipedia article on Cancellative Semigroups).

In the paragraph "Explicit Construction", I honestly do not see how the +k in the definition of the equivalence relation would imply that the monoid M is cancellative.

Moreover in this thread in Physics Forums, a user gives a simple example of an explicit construction of Grothendieck group for an abelian non-cancellative monoid. The resulting group obviously does not embed the given monoid.

--Bart.karviainen (talk) 13:14, 25 May 2010 (UTC)[reply]

"Best Possible Way"?

The first sentence of the article is vague and not rigorous. It says: "the grothendieck construction...constructs an abelian group from a commutative monoid, in the best possible way". It is definitely unclear what "best" means in this case. --Bart.karviainen (talk) 13:34, 25 May 2010 (UTC)[reply]

This is pretty rigorous and certainly not vague. And it's clear what "best" means in this context. Svycehbm (talk) 02:06, 18 June 2011 (UTC)[reply]

btw. "Best possible way" makes reference to the universal property. Maybe it deserves its own article(or section at Universal property). See another example usage inside wikipedia at Sheaf (mathematics) Svycehbm (talk) 03:08, 18 June 2011 (UTC)[reply]

Motivation

There is a clarify-tag for "zero element", and perhaps it is time to clarify it. Commutative monoid operation is quite often denoted by +, and accordingly, its neutral element by 0, which is what most people would instantly think of as "the zero element". Would "absorbing element" be less confusing, and much less unmotivating, or am I missing something? Lapasotka (talk) 13:15, 6 May 2022 (UTC)[reply]

"Absorbing element" would be correct, but too technical for a motivation section. I have rewritten the section for fixing this and other issues. D.Lazard (talk) 14:12, 6 May 2022 (UTC)[reply]