This chapter describes functions for transformations.
A transformation in GAP is an endomorphism of a set of integers of the form {1,..., n}. Transformations are taken to act on the right, which defines the composition i(ab) = (ia)b for i in {1, ..., n}.
For a transformation a on the set {1, ¼, n}, we define its degree to be n, its image list to be the list, [1a, ¼, na], its image to be the image list considered as a set, and its rank to be the size of the image. We also define the kernel of a to be the equivalence relation containing the pair (i, j) if and only if ia = ja.
Note that unlike permutations, we do not consider unspecified points to be fixed by a transformation. Therefore multiplication is only defined on two transformations of the same degree.
IsTransformation(
obj ) C
IsTransformationCollection(
obj ) C
We declare it as IsMultiplicativeElementWithOne since the identity automorphism of {1 .. n} is a multiplicative two sided identity for any transformation on the same set.
TransformationFamily( n ) F
TransformationType( n ) F
TransformationData( n ) F
For each n
> 0
there is a single family and type of transformations
on n points. To speed things up, we store these in
a database of types. The three functions above a then
access functions. If the nth entry isn't yet created, they trigger
creation as well.
For n
> 0
, element n of th etype database is
[TransformationFamily(n), TransformationType(n)]
Transformation(
images ) F
TransformationNC(
images ) F
both return a transformation with the image list images. The normal version checks that the all the elements of the given list lie within the range {1,...,n} where n is the length of images, but for speed purposes, a non-checking version is also supplied.
DegreeOfTransformation(
trans ) A
returns the degree of trans.
gap> t:= Transformation([2, 3, 4, 2, 4]); Transformation( [ 2, 3, 4, 2, 4 ] ) gap> DegreeOfTransformation(t); 5
ImageListOfTransformation(
trans ) A
returns the image list of trans.
gap> ImageListOfTransformation(t); [ 2, 3, 4, 2, 4 ]
ImageSetOfTransformation(
trans ) A
returns the image of trans as a set.
gap> ImageSetOfTransformation(t); [ 2, 3, 4 ]
RankOfTransformation(
trans ) A
returns the rank of trans.
gap> RankOfTransformation(t); 3
KernelOfTransformation(
trans ) A
Returns the kernel of trans as an equivalence relation (See General Binary Relations).
gap> KernelOfTransformation(t); <equivalence relation on [ 1 .. 5 ] > gap> EquivalenceRelationPartition(last); [ [ 1, 4 ], [ 3, 5 ] ]
PreimagesOfTransformation(
trans,
i ) O
returns the subset of {1,...,n} which maps to i under trans.
gap> PreimagesOfTransformation(t, 2); [ 1, 4 ]
AsTransformation(
O ) O
AsTransformation(
O ) O
AsTransformationNC(
O,
n ) O
returns the object O when considered as a transformation. In the second form, it returns O as a transformation of degree n, signalling an error if such a representation is not possible. AsTransformationNC does not perform this check.
gap> AsTransformation((1, 3)(2, 4)); Transformation( [ 3, 4, 1, 2 ] ) gap> AsTransformation((1, 3)(2, 4), 10); Transformation( [ 3, 4, 1, 2, 5, 6, 7, 8, 9, 10 ] ) gap> AsTransformation((1, 3)(2, 4), 3); Error Permutation moves points over the degree specified
BinaryRelationTransformation(
trans ) O
returns trans when considered as a binary relation.
TransformationRelation(
R ) O
returns the binary relation R when considered as a transformation. Only makes sense for injective binary relations over [1..n], Returns an error if the relation is not over [1..n], and fail if it is not injective.
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GAP 4 manual