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If I start with a commutative rig, and apply the Grothendieck Group construction to it, twice, once to the additive structure and once to the multiplicative structure, is the result well-known? Does the order of application of the construction matter?

In particular, starting from the semiring $\mathbb{N}$ of the naturals, what do I get for the multiplicative structure? It can't be quite isomorphic to the field $\mathbb{Q}$, but could it be $\mathbb{Q}$ endowed with a meadow structure [link is to a pdf]?

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What is a commutative rig? –  Bugs Bunny Aug 31 '12 at 10:14
A ring with no negatives. –  Jacques Carette Aug 31 '12 at 12:07
Just one thought although this seems not to be your situation and Grothendieck group definition you link to requires a neutral element: if your naturals where to not include 0, I think the order in which you do the 'quotient constructions' would make a difference. With multiplication first, you'd get the positive rationals and then the rationals. While with addition first you'd get the intgers and then just 0. –  quid Aug 31 '12 at 13:37
@quid: good point. Glad that my thought about application order being an issue was not completely off. –  Jacques Carette Aug 31 '12 at 20:27
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2 Answers

up vote 4 down vote accepted

The Grothendieck group of a semiring is a ring. So you are just asking what will happen if you take the multiplicative Grothendieck group of a ring.

Well, it depends on whether you include $0$ in the multiplicative structure! If you include $0$, then $a/b=0a/0b=0/0$ so you get the trivial group. Similarly, inverting zero-divisors will collapse part of the ring. If you only take the Grothendieck group of the multiplicative semigroup of non-zero-divisors, you get the unit group of the total quotient ring. In particular, for an integral domain you get the multiplicative group of the field of fractions. This construction is clearly the same as localization by whatever subset you pick.

If the original semiring is a subsemiring of a field, then it does not matter what order you pick. This is because you can do all the arithmetic inside the field, so the additive Grothendieck group is just the subset generated by a semiring under addition, multiplication and subtraction, and the multiplicative Grothendieck group is the subset generated under addition, multiplication, and division. Applying both gets the field generated by that semiring, regardless of order.

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I do think this indeed answers my question -- although please see the edits I made (to rig from semiring). Though I don't think this makes a difference. –  Jacques Carette Aug 30 '12 at 21:01
According to wikipedia, a rig is another word for semiring except that a semiring is sometimes not taken to have $1$ or $0$, so I don't think it makes a difference. –  Will Sawin Aug 30 '12 at 21:09
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  1. The Grothendieck group of a commutative group is group itself, hence applying the construction twice doesn't change the result. The semiring case is a special case of this.

  2. What do you mean by "order of application". There is no order if you apply the same thing twice. G(G(M)) is the same as G(G(M)) ...

  3. The Grothendieck group of the naturals is the ring of integers. This is the standard construction of the integers.

This sounds somewhat to trivial ... May be you mean something different here. Maybe you want to apply the construction to the multiplicative monoid instead of the additive structure in one of the two steps? The result in this case is (independent of the order) the zero ring because there is an absorbing element in the multiplicative monoid (the zero element of the semiring) so that all elements of the monoid get identified in the Grothendieck group.

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I will clarify my question: you are right, I meant to apply it to each monoidal structure once, not just the additive structure. –  Jacques Carette Aug 30 '12 at 20:38
And I meant commutative rig, not semiring, sorry. Though I was rather suspecting the last result (i.e. one gets the zero ring). I was just hoping that it wasn't the case. –  Jacques Carette Aug 30 '12 at 20:43
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