def __init__(self, M, prefix='T', category=None, **options):
r"""
Initialize ``self``.
EXAMPLES::
sage: C = CombinatorialFreeModule(QQ, ['a','b','c'])
sage: TA = TensorAlgebra(C)
sage: TestSuite(TA).run()
sage: m = SymmetricFunctions(QQ).m()
sage: Tm = TensorAlgebra(m)
sage: TestSuite(Tm).run()
"""
self._base_module = M
R = M.base_ring()
category = GradedHopfAlgebrasWithBasis(
R.category()).or_subcategory(category)
CombinatorialFreeModule.__init__(self,
R,
IndexedFreeMonoid(M.indices()),
prefix=prefix,
category=category,
**options)
# the following is not the best option, but it's better than nothing.
self._print_options['tensor_symbol'] = options.get(
'tensor_symbol', tensor.symbol)
def __classcall_private__(cls, index_set=None, names=None,
commutative=False, **kwds):
r"""
Construct a free monoid or a free abelian monoid, depending on the
input. Also, normalize the input.
EXAMPLES::
sage: F.<a,b,c,d,e> = FreeMonoid(); F
Free monoid on 5 generators (a, b, c, d, e)
sage: FreeMonoid(index_set=ZZ)
Free monoid indexed by Integer Ring
sage: F.<x,y,z> = FreeMonoid(abelian=True); F
Free abelian monoid on 3 generators (x, y, z)
sage: FreeMonoid(index_set=ZZ, commutative=True)
Free abelian monoid indexed by Integer Ring
sage: F = FreeMonoid(index_set=ZZ, names='x,y,z')
sage: G = FreeMonoid(index_set=ZZ, names=['x', 'y', 'z'])
sage: F == G
True
sage: F is G
True
sage: FreeMonoid(2, names='a,b') is FreeMonoid(names=['a','b'])
True
Fix a bug when ``index_set`` is ``None`` and ``names`` is a
string (:trac:`26221`)::
sage: FreeMonoid(2, names=['a','b']) is FreeMonoid(names='a,b')
True
"""
if 'abelian' in kwds:
commutative = kwds.pop('abelian')
if commutative:
from sage.monoids.free_abelian_monoid import FreeAbelianMonoid
return FreeAbelianMonoid(index_set, names, **kwds)
# Swap args (this works if names is None as well)
if isinstance(index_set, str):
names, index_set = index_set, names
if index_set is None and names is not None:
if isinstance(names, str):
index_set = names.count(',') + 1
else:
index_set = len(names)
if index_set not in ZZ:
if names is not None:
names = normalize_names(-1, names)
from sage.monoids.indexed_free_monoid import IndexedFreeMonoid
return IndexedFreeMonoid(index_set, names=names, **kwds)
if names is None:
raise ValueError("names must be specified")
names = normalize_names(index_set, names)
return super(FreeMonoid, cls).__classcall__(cls, index_set, names)
def free(index_set=None, names=None, **kwds):
r"""
Return a free monoid on `n` generators or with the generators
indexed by a set `I`.
A free monoid is constructed by specifing either:
- the number of generators and/or the names of the generators
- the indexing set for the generators
INPUT:
- ``index_set`` -- (optional) an index set for the generators; if
an integer, then this represents `\{0, 1, \ldots, n-1\}`
- ``names`` -- a string or list/tuple/iterable of strings
(default: ``'x'``); the generator names or name prefix
EXAMPLES::
sage: Monoids.free(index_set=ZZ)
Free monoid indexed by Integer Ring
sage: Monoids().free(ZZ)
Free monoid indexed by Integer Ring
sage: F.<x,y,z> = Monoids().free(); F
Free monoid indexed by {'x', 'y', 'z'}
"""
if names is not None:
if isinstance(names, str):
from sage.rings.all import ZZ
if ',' not in names and index_set in ZZ:
names = [names + repr(i) for i in range(index_set)]
else:
names = names.split(',')
names = tuple(names)
if index_set is None:
index_set = names
from sage.monoids.indexed_free_monoid import IndexedFreeMonoid
return IndexedFreeMonoid(index_set, names=names, **kwds)
def FreeMonoid(index_set=None, names=None, commutative=False, **kwds):
r"""
Return a free monoid on `n` generators or with the generators indexed by
a set `I`.
We construct free monoids by specifing either:
- the number of generators and/or the names of the generators
- the indexing set for the generators
INPUT:
- ``index_set`` -- an indexing set for the generators; if an integer,
than this becomes `\{0, 1, \ldots, n-1\}`
- ``names`` -- names of generators
- ``commutative`` -- (default: ``False``) whether the free monoid is
commutative or not
OUTPUT:
A free monoid.
EXAMPLES::
sage: F.<a,b,c,d,e> = FreeMonoid(); F
Free monoid on 5 generators (a, b, c, d, e)
sage: FreeMonoid(index_set=ZZ)
Free monoid indexed by Integer Ring
sage: F.<x,y,z> = FreeMonoid(abelian=True); F
Free abelian monoid on 3 generators (x, y, z)
sage: FreeMonoid(index_set=ZZ, commutative=True)
Free abelian monoid indexed by Integer Ring
TESTS::
sage: FreeMonoid(index_set=ZZ, names='x,y,z')
Free monoid indexed by Integer Ring
"""
if 'abelian' in kwds:
commutative = kwds.pop('abelian')
if commutative:
from sage.monoids.free_abelian_monoid import FreeAbelianMonoid
return FreeAbelianMonoid(index_set, names, **kwds)
if isinstance(index_set, str): # Swap args (this works if names is None as well)
names, index_set = index_set, names
if index_set is None and names is not None:
if isinstance(names, str):
index_set = names.count(',')
else:
index_set = len(names)
if index_set not in ZZ:
if names is not None:
names = normalize_names(-1, names)
from sage.monoids.indexed_free_monoid import IndexedFreeMonoid
return IndexedFreeMonoid(index_set, names=names, **kwds)
if names is None:
raise ValueError("names must be specified")
return FreeMonoid_factory(index_set, names)