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

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]
class sage.monoids.free_abelian_monoid.FreeAbelianMonoidFactory

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
sage: loads(dumps(F)) is F
True
__weakref__
list of weak references to the object (if defined)
create_key(n, names)
create_object(version, key)
class sage.monoids.free_abelian_monoid.FreeAbelianMonoid_class(n, names)

Free abelian monoid on n generators.

__call__(x)

Create an element of this abelian monoid from x.

EXAMPLES:

sage: F = FreeAbelianMonoid(10,'x')
sage: F(F.gen(2))
x2
sage: F(1)
1
__contains__(x)

Return True if x is an element of this abelian monoid.

EXAMPLES:

sage: F = FreeAbelianMonoid(10,'b')
sage: F.gen(2)*F.gen(3) in F
True

Note that a monoid on 9 generators is not considered a submonoid of one on 10 generators.

sage: FreeAbelianMonoid(9,'c').gen(2) in F
False

However, multiple calls to the monoid constructor do not return multiple distinct monoids.

sage: FreeAbelianMonoid(10,'b').gen(2) in F
True
__init__(n, names)
__repr__()
gen(i=0)

The i-th generator of the abelian monoid.

EXAMPLES:

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

The number of free generators of the abelian monoid.

EXAMPLES:

sage: F = FreeAbelianMonoid(3000, 'a')
sage: F.ngens()
3000
sage.monoids.free_abelian_monoid.is_FreeAbelianMonoid(x)

Return True if x is a free abelian monoid.

EXAMPLES:

sage: from sage.monoids.free_abelian_monoid import is_FreeAbelianMonoid
sage: is_FreeAbelianMonoid(5)
False
sage: is_FreeAbelianMonoid(FreeAbelianMonoid(7,'a'))
True
sage: is_FreeAbelianMonoid(FreeMonoid(7,'a'))
False
sage: is_FreeAbelianMonoid(FreeMonoid(0,''))
False

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