Free Monoids

AUTHORS:

  • David Kohel (2005-09)
  • Simon King (2011-04): Put free monoids into the category framework

Sage supports free monoids on any prescribed finite number \(n\geq 0\) 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.

sage.monoids.free_monoid.FreeMonoid(index_set=None, names=None, commutative=False, **kwds)

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

We construct free monoids by specifing either:

  • the number of generators and/or the names of the generators
  • the indexing set for the generators

INPUT:

  • index_set – an indexing set for the generators; if an integer, than this becomes \(\{0, 1, \ldots, n-1\}\)
  • names – names of generators
  • commutative – (default: False) whether the free monoid is commutative or not

OUTPUT:

A free monoid.

EXAMPLES:

sage: F.<a,b,c,d,e> = FreeMonoid(); F
Free monoid on 5 generators (a, b, c, d, e)
sage: FreeMonoid(index_set=ZZ)
Free monoid indexed by Integer Ring

sage: F.<x,y,z> = FreeMonoid(abelian=True); F
Free abelian monoid on 3 generators (x, y, z)
sage: FreeMonoid(index_set=ZZ, commutative=True)
Free abelian monoid indexed by Integer Ring
class sage.monoids.free_monoid.FreeMonoidFactory

Bases: sage.structure.factory.UniqueFactory

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

INPUT:

  • n - integer
  • names - names of generators

OUTPUT: free 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
create_key(n, names)

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

create_object(version, key, **kwds)

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

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

Bases: sage.monoids.monoid.Monoid_class

The free monoid on \(n\) generators.

Element

alias of FreeMonoidElement

cardinality()

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

EXAMPLES:

sage: F = FreeMonoid(2005, 'a')
sage: F.cardinality()
+Infinity
gen(i=0)

The \(i\)-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)
Traceback (most recent call last):
...
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
one_element()

Returns the identity element in this monoid.

EXAMPLES:

sage: F = FreeMonoid(2005, 'a')
sage: F.one_element()
1
sage.monoids.free_monoid.is_FreeMonoid(x)

Return True if \(x\) 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
sage: is_FreeMonoid(FreeMonoid(index_set=ZZ))
True
sage: is_FreeMonoid(FreeAbelianMonoid(index_set=ZZ))
False

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