Free Monoids

AUTHORS:

• David Kohel (2005-09)

Sage supports free monoids on any prescribed finite number of generators. Use the FreeMonoid function to create a free 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 FreeMonoid function.

class sage.monoids.free_monoid.FreeMonoidFactory

Returns a free monoid on generators.

INPUT:

• n - integer
• names - names of generators

OUTPUT: free abelian monoid

EXAMPLES:

sage: FreeMonoid(0,'')
Free monoid on 0 generators ()
sage: F.<a,b,c,d,e> = FreeMonoid(5); F
Free monoid on 5 generators (a, b, c, d, e)
sage: F(1)
1
sage: mul([ a, b, a, c, b, d, c, d ], F(1))
a*b*a*c*b*d*c*d

__weakref__

list of weak references to the object (if defined)

create_key(n, names)

create_object(version, key)

class sage.monoids.free_monoid.FreeMonoid_class(n, names=None)

The free monoid on generators.

__call__(x, check=True)

Return coerced into this free monoid.

One can create a free monoid from the integer 1 and from a list of 2-tuples of integers , where corresponds to , where is the `i`th generator.

EXAMPLES:

sage: F = FreeMonoid(3, 'a')
sage: F(1)
1
sage: F(F.gen(0))
a0
sage: F(0)
...
TypeError: Argument x (= 0) is of the wrong type.

sage: F([(0,5),(1,2),(0,10),(0,2),(1,2)])
a0^5*a1^2*a0^12*a1^2

__cmp__(other)

__contains__(x)

__init__(n, names=None)

Create free monoid on generators.

INPUT:

• n - integer
• names - (optional) variable name or list of variable names

EXAMPLES:

sage: F = FreeMonoid(3,'x'); F
Free monoid on 3 generators (x0, x1, x2)
sage: x = F.gens()
sage: x[0]*x[1]**5 * (x[0]*x[2])
x0*x1^5*x0*x2
sage: F = FreeMonoid(3, 'a')
sage: F
Free monoid on 3 generators (a0, a1, a2)

sage: M = FreeMonoid(3, names=['a','b','c'])
sage: loads(M.dumps()) == M
True

__repr__()

gen(i=0)

The -th generator of the monoid.

INPUT:

• i - integer (default: 0)

EXAMPLES:

sage: F = FreeMonoid(3, 'a')
sage: F.gen(1)
a1
sage: F.gen(2)
a2
sage: F.gen(5)
...
IndexError: Argument i (= 5) must be between 0 and 2.

ngens()

The number of free generators of the monoid.

EXAMPLES:

sage: F = FreeMonoid(2005, 'a')
sage: F.ngens()
2005

sage.monoids.free_monoid.is_FreeMonoid(x)

Return True if is a free monoid.

EXAMPLES:

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