Free abelian monoids#

AUTHORS:

  • David Kohel (2005-09)

Sage supports free abelian monoids on any prescribed finite number \(n\geq 0\) of generators. Use the FreeAbelianMonoid function to create a free abelian monoid, and the gen and gens functions to obtain the corresponding generators. You can print the generators as arbitrary strings using the optional names argument to the FreeAbelianMonoid function.

EXAMPLE 1: It is possible to create an abelian monoid in zero or more variables; the syntax T(1) creates the monoid identity element even in the rank zero case.

sage: T = FreeAbelianMonoid(0, '')
sage: T
Free abelian monoid on 0 generators ()
sage: T.gens()
()
sage: T(1)
1

>>> from sage.all import *
>>> T = FreeAbelianMonoid(Integer(0), '')
>>> T
Free abelian monoid on 0 generators ()
>>> T.gens()
()
>>> T(Integer(1))
1

EXAMPLE 2: A free abelian monoid uses a multiplicative representation of elements, but the underlying representation is lists of integer exponents.

sage: F = FreeAbelianMonoid(5,names='a,b,c,d,e')
sage: (a,b,c,d,e) = F.gens()
sage: a*b^2*e*d
a*b^2*d*e
sage: x = b^2*e*d*a^7
sage: x
a^7*b^2*d*e
sage: x.list()
[7, 2, 0, 1, 1]

>>> from sage.all import *
>>> F = FreeAbelianMonoid(Integer(5),names='a,b,c,d,e')
>>> (a,b,c,d,e) = F.gens()
>>> a*b**Integer(2)*e*d
a*b^2*d*e
>>> x = b**Integer(2)*e*d*a**Integer(7)
>>> x
a^7*b^2*d*e
>>> x.list()
[7, 2, 0, 1, 1]
sage.monoids.free_abelian_monoid.FreeAbelianMonoid(index_set=None, names=None, **kwds)[source]#

Return a free abelian monoid on \(n\) generators or with the generators indexed by a set \(I\).

We construct free abelian monoids by specifing either:

  • the number of generators and/or the names of the generators

  • the indexing set for the generators (this ignores the other two inputs)

INPUT:

  • index_set – an indexing set for the generators; if an integer, then this becomes \(\{0, 1, \ldots, n-1\}\)

  • names – names of generators

OUTPUT:

A free abelian monoid.

EXAMPLES:

sage: F.<a,b,c,d,e> = FreeAbelianMonoid(); F
Free abelian monoid on 5 generators (a, b, c, d, e)
sage: FreeAbelianMonoid(index_set=ZZ)
Free abelian monoid indexed by Integer Ring
sage: FreeAbelianMonoid(names='x,y')
Free abelian monoid on 2 generators (x, y)

>>> from sage.all import *
>>> F = FreeAbelianMonoid(names=('a', 'b', 'c', 'd', 'e',)); (a, b, c, d, e,) = F._first_ngens(5); F
Free abelian monoid on 5 generators (a, b, c, d, e)
>>> FreeAbelianMonoid(index_set=ZZ)
Free abelian monoid indexed by Integer Ring
>>> FreeAbelianMonoid(names='x,y')
Free abelian monoid on 2 generators (x, y)
class sage.monoids.free_abelian_monoid.FreeAbelianMonoidFactory[source]#

Bases: UniqueFactory

Create the free abelian monoid in \(n\) generators.

INPUT:

  • n – integer

  • names – names of generators

OUTPUT: free abelian monoid

EXAMPLES:

sage: FreeAbelianMonoid(0, '')
Free abelian monoid on 0 generators ()
sage: F = FreeAbelianMonoid(5,names = list("abcde"))
sage: F
Free abelian monoid on 5 generators (a, b, c, d, e)
sage: F(1)
1
sage: (a, b, c, d, e) = F.gens()
sage: mul([ a, b, a, c, b, d, c, d ], F(1))
a^2*b^2*c^2*d^2
sage: a**2 * b**3 * a**2 * b**4
a^4*b^7

>>> from sage.all import *
>>> FreeAbelianMonoid(Integer(0), '')
Free abelian monoid on 0 generators ()
>>> F = FreeAbelianMonoid(Integer(5),names = list("abcde"))
>>> F
Free abelian monoid on 5 generators (a, b, c, d, e)
>>> F(Integer(1))
1
>>> (a, b, c, d, e) = F.gens()
>>> mul([ a, b, a, c, b, d, c, d ], F(Integer(1)))
a^2*b^2*c^2*d^2
>>> a**Integer(2) * b**Integer(3) * a**Integer(2) * b**Integer(4)
a^4*b^7

sage: loads(dumps(F)) is F
True

>>> from sage.all import *
>>> loads(dumps(F)) is F
True
create_key(n, names)[source]#
create_object(version, key)[source]#
class sage.monoids.free_abelian_monoid.FreeAbelianMonoid_class(n, names)[source]#

Bases: Parent

Free abelian monoid on \(n\) generators.

Element[source]#

alias of FreeAbelianMonoidElement

cardinality()[source]#

Return the cardinality of self, which is \(\infty\).

EXAMPLES:

sage: F = FreeAbelianMonoid(3000, 'a')
sage: F.cardinality()
+Infinity

>>> from sage.all import *
>>> F = FreeAbelianMonoid(Integer(3000), 'a')
>>> F.cardinality()
+Infinity
gen(i=0)[source]#

The \(i\)-th generator of the abelian monoid.

EXAMPLES:

sage: F = FreeAbelianMonoid(5,'a')
sage: F.gen(0)
a0
sage: F.gen(2)
a2

>>> from sage.all import *
>>> F = FreeAbelianMonoid(Integer(5),'a')
>>> F.gen(Integer(0))
a0
>>> F.gen(Integer(2))
a2
gens()[source]#

Return the generators of self.

EXAMPLES:

sage: F = FreeAbelianMonoid(5,'a')
sage: F.gens()
(a0, a1, a2, a3, a4)

>>> from sage.all import *
>>> F = FreeAbelianMonoid(Integer(5),'a')
>>> F.gens()
(a0, a1, a2, a3, a4)
ngens()[source]#

The number of free generators of the abelian monoid.

EXAMPLES:

sage: F = FreeAbelianMonoid(3000, 'a')
sage: F.ngens()
3000

>>> from sage.all import *
>>> F = FreeAbelianMonoid(Integer(3000), 'a')
>>> F.ngens()
3000
sage.monoids.free_abelian_monoid.is_FreeAbelianMonoid(x)[source]#

Return True if \(x\) is a free abelian monoid.

EXAMPLES:

sage: from sage.monoids.free_abelian_monoid import is_FreeAbelianMonoid
sage: is_FreeAbelianMonoid(5)
doctest:warning...
DeprecationWarning: the function is_FreeAbelianMonoid is deprecated;
use 'isinstance(..., FreeAbelianMonoid_class)' instead
See https://github.com/sagemath/sage/issues/37897 for details.
False
sage: is_FreeAbelianMonoid(FreeAbelianMonoid(7,'a'))
True
sage: is_FreeAbelianMonoid(FreeMonoid(7,'a'))
False
sage: is_FreeAbelianMonoid(FreeMonoid(0,''))
False

>>> from sage.all import *
>>> from sage.monoids.free_abelian_monoid import is_FreeAbelianMonoid
>>> is_FreeAbelianMonoid(Integer(5))
doctest:warning...
DeprecationWarning: the function is_FreeAbelianMonoid is deprecated;
use 'isinstance(..., FreeAbelianMonoid_class)' instead
See https://github.com/sagemath/sage/issues/37897 for details.
False
>>> is_FreeAbelianMonoid(FreeAbelianMonoid(Integer(7),'a'))
True
>>> is_FreeAbelianMonoid(FreeMonoid(Integer(7),'a'))
False
>>> is_FreeAbelianMonoid(FreeMonoid(Integer(0),''))
False