# 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]

sage.monoids.free_abelian_monoid.FreeAbelianMonoid(index_set=None, names=None, **kwds)

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

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

create_key(n, names)

x.__init__(...) initializes x; see help(type(x)) for signature

create_object(version, key)

x.__init__(...) initializes x; see help(type(x)) for signature

class sage.monoids.free_abelian_monoid.FreeAbelianMonoid_class(n, names)

Free abelian monoid on $$n$$ generators.

cardinality()

Return the cardinality of self, which is $$\infty$$.

EXAMPLES:

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

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


Monoid Elements

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Abelian monoid elements