Timeline for Standard method and name for extending a semiring to a ring
Current License: CC BY-SA 3.0
5 events
| when toggle format | what | by | license | comment | |
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| Mar 8, 2012 at 19:59 | comment | added | Benjamin Steinberg | You can construct the Grothendieck group of the additive monoid via a fraction style construction (except the operation is addition). Then the multiplication can be extended in a natural way. Of course if addition is idempotent then you will collapse the whole semiring. | |
| Mar 8, 2012 at 13:32 | comment | added | Richard | I can't really understand much in the book you referenced, and I can't see where it defines multiplication over the Cartesian product, but I have defined it distributively (p1,n1)(p2,n2)=(p1 p2 + n1 n2, p1 n2 + n1 p2) I've realized that an optimal solution defined by the Grothendieck group is necessary, otherwise inverses are not unique = no ring. Max-plus semi-rings under this construction collapse to a single element (as idempotent operations can't have inverses). I guess a quotient creating a trivial ring is always guaranteed to create a ring!! Though not what I was hoping for. | |
| Mar 8, 2012 at 8:33 | vote | accept | Richard | ||
| Mar 8, 2012 at 8:33 | comment | added | Richard | That is exactly what I was looking for. The standard construction for Grothendieck rings is very similar to the above method of localization - using a Cartesian product and defining new operations - and I had in fact considered something like it. I'm not sure if my proposed construction is isomorphic or not, I suspect that my equivalence relation is suboptimal (the newly constucted group will be larger than necessary), though it still should work. | |
| Mar 8, 2012 at 4:16 | history | answered | Benjamin Steinberg | CC BY-SA 3.0 |