Daniel Fischer wrote:
> Am Mittwoch 07 Oktober 2009 23:51:54 schrieb Joe Fredette: > > I generally find semirings defined as a ring > > structure without additive inverse and with 0-annihilation (which one > > has to assume in the case of SRs, I included it in my previous > > definition because I wasn't sure if I could prove it via the axioms, I > > think it's possible, but I don't recall the proof). > > 0*x = (0+0)*x = 0*x + 0*x ==> 0*x = 0 This proof only works if your additive monoid is cancellative, which need not be true in a semiring. The natural numbers extended with infinity is one example (if you don't take 0*x = 0 as an axiom, I think there are two possibilities for 0*∞). -- Jason McCarty <[hidden email]> _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
On Wed, Oct 07, 2009 at 08:44:27PM -0400, Jason McCarty wrote:
> Daniel Fischer wrote: > > Am Mittwoch 07 Oktober 2009 23:51:54 schrieb Joe Fredette: > > > I generally find semirings defined as a ring > > > structure without additive inverse and with 0-annihilation (which one > > > has to assume in the case of SRs, I included it in my previous > > > definition because I wasn't sure if I could prove it via the axioms, I > > > think it's possible, but I don't recall the proof). > > > > 0*x = (0+0)*x = 0*x + 0*x ==> 0*x = 0 > > This proof only works if your additive monoid is cancellative, which > need not be true in a semiring. The natural numbers extended with > infinity is one example (if you don't take 0*x = 0 as an axiom, I think > there are two possibilities for 0*∞). Given that x = 1*x = (0+1)*x = 0*x + 1*x = 0*x + x we can show that x = x + 0*x (right) x = 0*x + x (left) so, by definition of 'zero', we have that 0*x is a zero. But we can easily prove that there can be only one zero: suppose we have two zeros z1 and z2; it follows that z1 = z1 + z2 = z2 So 0*x = 0. Any flaws? -- Felipe. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Am Donnerstag 08 Oktober 2009 03:05:13 schrieb Felipe Lessa:
> On Wed, Oct 07, 2009 at 08:44:27PM -0400, Jason McCarty wrote: > > Daniel Fischer wrote: > > > Am Mittwoch 07 Oktober 2009 23:51:54 schrieb Joe Fredette: > > > > I generally find semirings defined as a ring > > > > structure without additive inverse and with 0-annihilation (which one > > > > has to assume in the case of SRs, I included it in my previous > > > > definition because I wasn't sure if I could prove it via the axioms, > > > > I think it's possible, but I don't recall the proof). > > > > > > 0*x = (0+0)*x = 0*x + 0*x ==> 0*x = 0 > > > > This proof only works if your additive monoid is cancellative, which > > need not be true in a semiring. The natural numbers extended with > > infinity is one example (if you don't take 0*x = 0 as an axiom, I think > > there are two possibilities for 0*∞). It was a proof for a ring (with or without unit), which Joe stated above he didn't recall. There your additive monoid is cancellative since it's a group :D > > Given that > > x = 1*x = (0+1)*x = 0*x + 1*x = 0*x + x > > we can show that > > x = x + 0*x (right) > x = 0*x + x (left) > > so, by definition of 'zero', we have that 0*x is a zero. Not necessarily, we don't know 0*x + y = y for arbitrary y yet. If the additive monoid isn't cancellative, that needn't be the case. In Jason's example, you can indeed set 0*∞ = ∞. > But we can easily prove that there can be only one zero: suppose we have > two zeros z1 and z2; it follows that > > z1 = z1 + z2 = z2 > > So 0*x = 0. Any flaws? > > -- > Felipe. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
In reply to this post by jfredett
Joe Fredette wrote:
> A ring is an abelian group in addition, with the added operation (*) > being distributive over addition, and 0 annihilating under > multiplication. (*) is also associative. Rings don't necessarily need > _multiplicative_ id, only _additive_ id. Yes, this is how I learned it in my Algebra course(*). Though I can imagine that this is not universally agreed upon; indeed most of the more interesting results need a multiplicative unit, IIRC, so there's a justification for authors to include it in the basic definition (so they don't have to say let-R-be-a-ring-with-multiplicative-unit all the time ;-) Cheers Ben (*) As an aside, this was given by one Gernot Stroth, back then still at the FU Berlin, of whom I later learned that he took part in the grand effort to classify the simple finite groups. The course was extremely tight but it was fun and I never again learned so much in one semester. _______________________________________________ Haskell-Cafe mailing list [hidden email] http://www.haskell.org/mailman/listinfo/haskell-cafe |
Free forum by Nabble | Edit this page |