Transf, PPerm, and Perm helpers

This page contains the documentation for various helper functions for manipulating transformations, partial permutations, and permutations.

Contents

`domain`(…)	Returns the set of points where a partial transformation is defined.
`image`(…)	Returns the set of points in the image of a partial transformation.
`inverse`(…)	Returns the inverse of a partial perm or permutation.
`left_one`(…)	Returns the left one of a partial perm.
`one`(…)	Returns the identity on the same number of points as the degree of f.
`right_one`(…)	Returns the right one of a partial perm.

Full API

This page contains the documentation for the transf subpackage, that contains helper functions for the Transf, Perm, PPerm classes.

transf.domain(f: Transf | PPerm | Perm) → list[int]

Returns the set of points where a partial transformation is defined.

This function returns a list[int] containing those values i such that:

\(i\in \{0, \ldots, n - 1\}\) where n is the degree of f; and
f[i] != UNDEFINED (only applies to partial perms).

Parameters:: f (Transf | PPerm | Perm) – the transformation, partial perm, or permutation.
Returns:: The sorted list of points in the domain.
Return type:: list[int]
Complexity:: Linear in the degree of f.

transf.image(f: Transf | PPerm | Perm) → list[int]

Returns the set of points in the image of a partial transformation.

Returns a list[int] containing those values f[i] such that:

\(i\in \{0, \ldots, n - 1\}\) where n is the degree of f; and
f[i] != UNDEFINED.

Parameters:: f (Transf | PPerm | Perm) – the transformation.
Returns:: The sorted list of points in the image.
Return type:: list[int]
Complexity:: Linear in the degree of f.

transf.inverse(f: PPerm | Perm) → PPerm | Perm

Returns the inverse of a partial perm or permutation.

This function returns a newly constructed inverse of f. The inverse of a partial permutation f is the partial perm g such that fgf = f and gfg = g.

Parameters:: f (PPerm | Perm) – the partial perm.
Returns:: The inverse of f.
Return type:: PPerm | Perm
Complexity:: Linear in the degree of f.

transf.left_one(f: PPerm) → PPerm

Returns the left one of a partial perm.

This function returns a newly constructed partial perm with degree equal to that of f that fixes every value in the domain of f, and is UNDEFINED on any other values.

Parameters:: f (PPerm) – the partial perm.
Returns:: The left one of f.
Return type:: PPerm
Complexity:: Linear in the degree of f.

transf.one(f: Transf | PPerm | Perm) → Transf | PPerm | Perm

Returns the identity on the same number of points as the degree of f. This function returns a newly constructed object of the same type as f that fixes every value from 0 to f.degree().

Parameters:: f (Transf | PPerm | Perm) – the transformation, partial perm, or permutation.
Returns:: The identity.
Return type:: Transf | PPerm | Perm
Complexity:: Linear in the degree of f.

transf.right_one(f: PPerm) → PPerm

Returns the right one of a partial perm.

This function returns a newly constructed partial perm with degree equal to that of f that fixes every value in the image of f, and is UNDEFINED on any other values.

Parameters:: f (PPerm) – the partial perm.
Returns:: The right one of f.
Return type:: PPerm
Complexity:: Linear in the degree of f.