Is there a special name for an operator monoid where the structure that's acted on is an Abelian group?
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."
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...)
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...)
Some variation on a module then: http://en.wikipedia.org/wiki/Module_(mathematics)
On 28 Oct 2008, at 3:51 pm, Derek Elkins wrote:
Some variation on a module then: http://en.wikipedia.org/wiki/Module_(mathematics)
When you have M acting on X and X is an abelian group: M is a field => vector space M is a ring => module M is a semiring => module over a semiring M is a monoid => ??? Maybe I can generalise M to a ring after all...
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.
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Derek Elkins -
Richard O'Keefe