[TYPES] What algebra am I thinking of?

[Harrison Brown]

Hi Phil,

I think Meola and Eugenio have it right, and in particular I think you're
describing the free module over N generated by the set of resources. (Free
modules over rings proper behave pretty much like vector spaces; I don't
know much about the theory of free modules over rigs, but the analogy
should hold.) And, analogous to how free abelian groups are free modules
over Z, I believe that this is isomorphic to a free commutative monoid
generated by the set of resources. If you want to forbid the empty
transaction, you have a free commutative semigroup. Of course if the
resources interact in some way you no longer have a free structure, but you
should still have an N-module.

Best,

Harrison

On Tue, Mar 27, 2018 at 7:38 AM, Eugenio Moggi <moggi at disi.unige.it> wrote:

> [ The Types Forum, http://lists.seas.upenn.edu/mailman/listinfo/types-list
> ]
>
> > Consider a blockchain managing several different resources. Over time,
> new
> > resources may be added or deleted. Each input to or output from a
> > transaction is associated with a value, where each value consists of
> > associating zero or more resources with amounts, where the amounts are
> > natural numbers (that is, integers greater than or equal to zero).
> >
> > What kind of algebra do values correspond to? It seems similar to vector
> > spaces, except:
> >   (a) adding or deleting resources increases or decreases the number of
> > dimensions in the vector space
> >   (b) the scalars in the vector space are natural numbers rather than
> reals
> >
> > What algebra am I thinking of? Cheers, -- P
>
> Dear Phil, regarding the issue (b) you want to replace the FIELD of the
> real
> numbers with the SEMI-RING of natural numbers (N,+,*,0,1).
>
> As suggested by Meola the structure should be a module over a RIG, see
> https://ncatlab.org/nlab/show/module
> More precisely
> - a rig (F,*,+,1,0) for the scalars
> - a commutative monoid (V,+,0) for the vectors
> - an action *:FxV->V satisfying certain properties, in particular 0*v=0=f*0
>
> When F is a field, one recovers the usual notion of vector space.
>
> A vector space V can be infinite dimensional. In the case of a module over
> a
> RIG, the definition of base B should be the usual one, namely a subset of
> B is a
> base for V iff
>
> - every finite subset of B is linearly independent
> - every element of V is the linear combination of a finite subset of B
>
> but the definition of LINEARLY INDEPENDENT has to be revised, to avoid the
> use
> of "negative".  A finite subset {v_i|i:n} of V is lineraly independent <=>
> for every a,b:F^n if Sum_i a_i*v_i = Sum_i b_i*v_i, then a=b.
>
> In the module of over N proposed by Meola, ie the maps from the set of
> resourses
> to the rig N that have finite support, the base is unique, and there is an
> obvious definition of inner product, but I doubt you can do much with it.
>
> However, there are modules over N that have no base.  For instance, the
> module Z
> over N, {1} is too small to be a base, and {1,-1} is too big.
>
> Best Regards
> Eugenio Moggi
>