On Tue, 2008-10-28 at 15:43 +1300, Richard O'Keefe wrote: > On 28 Oct 2008, at 2:54 pm, Derek Elkins wrote: > > > On Tue, 2008-10-28 at 13:54 +1300, Richard O'Keefe wrote: > >> Is there a special name for an operator monoid where the > >> structure that's acted on is an Abelian group? > > > > This should just be equivalent to a ring, maybe without > > distributivity. > > Maybe missing some other properties depending on what you mean by > > "operator." > > Yes, it's close to a ring, but we have ((M,*,1),(X,+,0,-)) > where (M,*,1) is the monoid and (X,+,0,-) is the Abelian group. > For what I have in mind the sets M and X are disjoint. > For a ring they would be identical. > (This being Haskell-Café, I knew types would come in useful...)
Actually modules more or less come back to rings. The issue again comes back to what you mean by "operator". Let's pick a general notion. You haven't provided enough information above because you've given no way to connect M and X, which I'll call G for clarity. So let's posit an operation, a monoid action, M x G -> G to have M operate on G. M -> End(G) is isomorphic (via an isomorphism we all know and love.) End(G) is at least a non-distributive "ring" with * = o, and + = + (pointwise). So for a given f : M -> End(G), f(M) ~ H \subset End(G). H is a submonoid of End(G) which we can extend with all "sums" of elements in H into a non-distributive "ring" that is a subset of End(G). If f(m) for all m in M does distribute over +, then the extension step is unnecessary and H is a ring. This is a bunch of random unchecked math. At any rate, I don't think there is a specific name for exactly what you want, though I'm still not sure quite what you want. _______________________________________________ Haskell-Cafe mailing list Haskell-Cafe@haskell.org http://www.haskell.org/mailman/listinfo/haskell-cafe