#17160: Finitely generated axiom for (mutiplicative) magmas, semigroups,
monoids,
groups
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Reporter: nthiery | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.4
Component: categories | Resolution:
Keywords: | Merged in:
Authors: Nicolas M. Thiéry | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/nthiery/categories/finitely- | f027ce2b5e1abe22d49bcdc96f2cfeebced8fc16
generated-magmas-17160 | Stopgaps:
Dependencies: #10668 |
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Comment (by nthiery):
Replying to [comment:11 tscrim]:
> I have things that have infinite (enumerable) distinguished generating
sets (ex. free group/monoid with generators indexed by `NN` or Yangians
#15484), so separating these axioms will be useful. In fact, the
enumeration could be done in for the general `WithGeneratingSet` category
and would (at least should) error out if the generating set is not
enumerable. Although I only know of 1 thing which will be finite
dimensional but doesn't come with a distinguished basis.
Another issue: having a distinguished set of generators and being
finitely generated does not necessarily imply that the distinguished
set of generators is finite. So we would actually need three axioms:
"WithGenerators", "FinitelyGenerated", and "WithFiniteGeneratingSet".
So for a finite magma we still would need to do
"Magmas().Finite().WithFiniteGeneratingSet()".
I am not sure this is worth the complication. Especially since we will
have to do something similar for additive magmas, rings, fields, ...
> Plus I think we could do an extra case in `_repr_object_names_static` to
change the repr into:
> {{{
> Category of groups with finite generating set
> }}}
That should be easy indeed.
> Here's another thought, what about we look at the cardinality of the
generating set? So we only have `WithGeneratingSet` which calls
`is_finitely_generated`, whose default is to look at the cardinality of
the generating set to determine the output of repr. At least that's the
only place where I could see us (currently) using the fact that the
generating set is finite.
Well, also all the code to build the Cayley graph, to compute
J/R/L-classes, etc. In short all my finite semigroups code :-)
I oppose querying the cardinality, or even just is_finite, for this
can be super expensive if not undecidable. We really want something
declarative here.
> Darij, Aladin, or anyone else, your thoughts?
Yup?
We probably should bring the discussion to sage-dev. As usual this
takes a bit of preparation to have an efficient discussion there.
I'll try to do this soon.
Cheers,
Nicolas
--
Ticket URL: <http://trac.sagemath.org/ticket/17160#comment:12>
Sage <http://www.sagemath.org>
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