Up One Level SAGE Reference Manual Contents Module Index Index
Previous: Up: 20. Monoids Next:

20.3 Free abelian monoids

Module: sage.monoids.free_abelian_monoid

Free abelian monoids

Author: 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 indentity 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]

Module-level Functions

FreeAbelianMonoid( n, names)
Create the free abelian monoid in $ n$ generators. 

INPUT:
    n -- integer
    names -- names of generators

OUTPUT:
    free abelian monoid


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

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

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

Class: FreeAbelianMonoid_class

class FreeAbelianMonoid_class
Free abelian monoid on $ n$ generators.
FreeAbelianMonoid_class( self, n, names)

Functions: gen,$ $ ngens

gen( self, [i=0])
The $ i$-th generator of the abelian monoid. 

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

ngens( self)
The number of free generators of the abelian monoid. 

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

Special Functions: __call__,$ $ __contains__,$ $ __repr__

__call__( self, x)
Create an element of this abelian monoid from $ x$. 

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

__contains__( self, x)
Return True if $ x$ is an element of this abelian monoid. 

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

Up One Level SAGE Reference Manual Contents Module Index Index
Previous: Up: 20. Monoids Next:
Release 2007.10.28, documentation updated on October 28, 2007.
See About this document... for information on suggesting changes.