16.1 Free algebras

Module: sage.algebras.free_algebra

Author: David Kohel, 2005-09

Module-level Functions

FreeAlgebra( R, n, [names=None])

Return the free algebra over the ring $ R$ on $ n$ generators with given names.

INPUT:
    R -- ring
    n -- integer
    names -- string or list/tuple of n strings

OUTPUT:
    a free algebra

sage: FreeAlgebra(GF(5),3)
Free Algebra on 3 generators (x0, x1, x2) over Finite Field of size 5
sage: FreeAlgebra(GF(5),3, ['xx', 'zba', 'Y'])
Free Algebra on 3 generators (xx, zba, Y) over Finite Field of size 5
sage: FreeAlgebra(GF(5),3, 'abc')
Free Algebra on 3 generators (a, b, c) over Finite Field of size 5
sage: FreeAlgebra(GF(5),1, 'z')
Free Algebra on 1 generators (z,) over Finite Field of size 5
sage: FreeAlgebra(GF(5),1, ['alpha'])
Free Algebra on 1 generators (alpha,) over Finite Field of size 5
sage: FreeAlgebra(FreeAlgebra(ZZ,1), 2)
Free Algebra on 2 generators (x0, x1) over Free Algebra on 1 generators
(x,) over Integer Ring

is_FreeAlgebra( x)

Class: FreeAlgebra_generic

class FreeAlgebra_generic
FreeAlgebra_generic( self, R, n, [names=None])

Returns the free algebra on $ n$ generators.

sage: F = FreeAlgebra(QQ,ZZ(3),names=("x","y","z"))
sage: mul([ F.gen(i) for i in range(3) ], F(1))
x*y*z
sage: mul([ F.gen(i%3) for i in range(12) ], F(1))
x*y*z*x*y*z*x*y*z*x*y*z
sage: (x,y,z) = F.gens()
sage: (2 + x*z + x**2)**2 + (x - y)**2
4 + 3*x^2 - x*y + 2*x*z - y*x + y^2 + x^4 + x^3*z + x*z*x^2 + x*z*x*z

Functions: assign_names,$  $ base_ring,$  $ gen,$  $ monoid,$  $ ngens

assign_names( self, names)

Assign the printing names for the generators; this will have the unfortunate effect of overwriting the names for the covering algebra; this also does not overwrite the return value of names() for the Algebra.

gen( self, i)

The i-th generator of the algebra.

monoid( self)

The free monoid of generators of the algebra.

ngens( self)

The number of generators of the algebra.

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

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