The Transf class
Class for representing transformations on up to 2 ** 32 points.
A transformation \(f\) is just a function defined on the whole of \(\{0, 1, \ldots, n - 1\}\) for some integer \(n\) called the degree of \(f\). A transformation is stored as a list of the images of \(\{0, 1, \ldots, n - 1\}\), i.e. \([(0)f, (1)f, \ldots, (n - 1)f]\).
Transformations are optimised for the number of points in the image with fewer points requiring less space per point.
>>> from libsemigroups_pybind11.transf import Transf, one
>>> x = Transf([0, 0, 2, 2, 0, 1])
>>> x.degree()
6
>>> x[0]
0
>>> x[5]
1
>>> x
Transf([0, 0, 2, 2, 0, 1])
>>> x * x
Transf([0, 0, 2, 2, 0, 0])
>>> x < x * x
False
>>> y = Transf([9, 7, 3, 5, 3, 4, 2, 7, 7, 1])
>>> x = one(y)
>>> x.product_inplace(y, y)
>>> x
Transf([1, 7, 5, 4, 5, 3, 3, 7, 7, 7])
>>> list(x.images())
[1, 7, 5, 4, 5, 3, 3, 7, 7, 7]
>>> x.rank()
5
>>> one(x)
Transf([0, 1, 2, 3, 4, 5, 6, 7, 8, 9])
>>> x = Transf.one(8)
>>> x
Transf([0, 1, 2, 3, 4, 5, 6, 7])
>>> x.degree()
8
>>> x.swap(y)
>>> x, y
(Transf([9, 7, 3, 5, 3, 4, 2, 7, 7, 1]), Transf([0, 1, 2, 3, 4, 5, 6, 7]))
>>> x = Transf([1, 0, 2])
>>> y = x.copy()
>>> x is y
False
>>> x == y
True
>>> {x, y}
{Transf([1, 0, 2])}
Contents
Class for representing transformations on up to |
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Copy a transformation. |
Returns an iterator to the images of a transformation. |
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Increases the degree of self in-place, leaving existing values unaltered. |
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Returns the identity transformation on N points. |
Replaces the contents of self by the product of x and y. |
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Returns the number of distinct image values in a transformation. |
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Swap with another transformation of the same type. |
Full API
- class Transf
- __init__(self: Transf, imgs: list[int]) None
A transformation can be constructed from a list of images, as follows: the image of the point
iunder the transformation isimgs[i].- Parameters:
- Raises:
LibsemigroupsError – if any value in imgs exceeds
len(imgs).- Complexity:
Linear in
degree().
- degree(self: Transf) int
Returns the degree of a transformation.
The degree of a transformation is the number of points used in its definition, which is equal to the size of
Transf.images.- Returns:
The degree.
- Return type:
- images(self: Transf) collections.abc.Iterator[int]
Returns an iterator to the images of a transformation.
A transformation is stored as a list of the images of \(\{0, 1, \ldots, n - 1\}\), i.e. \([(0)f, (1)f, \ldots, (n - 1)f]\), and this function returns an iterator yielding these values.
- Returns:
An iterator to the image values.
- Return type:
- increase_degree_by(self: Transf, m: int) Transf
Increases the degree of self in-place, leaving existing values unaltered.
- static one(N: int) Transf
Returns the identity transformation on N points. This function returns a newly constructed transformation with degree equal to N that fixes every value from
0to N.
- product_inplace(self: Transf, x: Transf, y: Transf) None
Replaces the contents of self by the product of x and y.