On Wed, Aug 06, 2014 at 03:44:26PM +0200, Daniel Krenn wrote:
> This sounds like a coercion from the monoid to the group, but IMHO
> should be the other way round. Each group element is also an monoid
> element (therefore the coercion always works).
+1. It might make sense to define the group as a submonoid of the
monoid using the Monoids.Subobjects infrastructure.
Also, if the elements of your monoid are functions from a finite set
to itself, you might want to represent them internally (e.g. using
ElementWrapper) as elements of a FiniteSetMaps. Here is an extract of
some experimental code doing things along this direction. It's not
functional without more stuff around, but this could give you an idea.
I won't have so much time in the coming semester (heavy teaching), but
we have a bunch of experimental monoid code lurking around that we
would like to get in Sage (see [1]). Maybe there is some opportunity
to join forces on this.
Cheers,
Nicolas
class SubFiniteMonoidsOfFunctions(Category):
@cached_method
def super_categories(self):
return [Monoids().Finite()]
class ParentMethods:
def domain(self):
return self.ambient().domain()
class ElementMethods:
def __call__(self, *args):
return self.lift()(*args)
def image_set(self):
return self.lift().image_set()
def rank(self):
return self.lift().rank()
def fibers(self):
return self.lift().fibers()
def domain(self):
return self.lift().domain()
def codomain(self):
return self.lift().codomain()
class HeckeMonoid(AutomaticMonoid):
"""
The 0-Hecke monoid of a Coxeter group
EXAMPLES::
sage: from sage.combinat.j_trivial_monoids import *
sage: H = HeckeMonoid(['A',3])
sage: pi = H.semigroup_generators(); pi
sage: pi
Finite family {1: [1], 2: [2], 3: [3]}
sage: H.cardinality()
24
sage: TestSuite(H).run()
"""
@staticmethod
def __classcall__(cls, W):
"""
EXAMPLES::
sage: from sage.combinat.j_trivial_monoids import *
sage: HeckeMonoid(['A',3]).cardinality()
24
"""
from sage.categories.coxeter_groups import CoxeterGroups
if not W in CoxeterGroups():
from sage.combinat.root_system.weyl_group import WeylGroup
W = WeylGroup(W) # CoxeterGroup(W)
return super(PiMonoid, cls).__classcall__(cls, W)
def __init__(self, W):
self.lattice = W.domain()
self.W = W
from sage.sets.finite_set_maps import FiniteSetMaps
ambient_monoid = FiniteSetMaps(self.W, action="right")
pi = self.W.simple_projections(length_increasing =
True).map(ambient_monoid)
AutomaticMonoid.__init__(self, pi, one = ambient_monoid.one(),
category = (SubFiniteMonoidsOfFunctions(),
JTrivialMonoids().Finite()))
self.domain = self.W
self.codomain = self.W
self.cache_element_methods(["rank"])
def _repr_(self):
return "zero_hecke monoid for type %s"%self.W.cartan_type()
Nicolas
PS:
[1] http://combinat.sagemath.org/patches/file/tip/finite_semigroup-nt.patch
(this address might need massaging; I don't have internet right now to
double check it).
--
Nicolas M. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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