The functions described in this section extend the functionality of GAP relating to transformations.
> TransformationByKernelAndImage( ker, img ) | ( operation ) |
> TransformationByKernelAndImageNC( ker, img ) | ( operation ) |
returns the transformation f with kernel ker and image img where (x)f=img[i] for all x in ker[i]. The argument ker should be a set of sets that partition the set 1,...n for some n and img should be a sublist of 1,...n.
TransformationByKernelAndImage first checks that ker and img describe the kernel and image of a transformation whereas TransformationByKernelAndImageNC performs no such check.
gap> TransformationByKernelAndImageNC([[1,2,3,4],[5,6,7],[8]],[1,2,8]); Transformation( [ 1, 1, 1, 1, 2, 2, 2, 8 ] ) gap> TransformationByKernelAndImageNC([[1,6],[2,5],[3,4]], [4,5,6]); Transformation( [ 4, 5, 6, 6, 5, 4 ] ) |
> AllTransformationsWithKerAndImg( ker, img ) | ( operation ) |
> AllTransformationsWithKerAndImgNC( ker, img ) | ( operation ) |
returns a list of all transformations with kernel ker and image img. The argument ker should be a set of sets that partition the set 1,...n for some n and img should be a sublist of 1,...n.
AllTransformationsWithKerAndImg first checks that ker and img describe the kernel and image of a transformation whereas AllTransformationsWithKerAndImgNC performs no such check.
gap> AllTransformationsWithKerAndImg([[1,6],[2,5],[3,4]], [4,5,6]);
[ Transformation( [ 4, 5, 6, 6, 5, 4 ] ),
Transformation( [ 6, 5, 4, 4, 5, 6 ] ),
Transformation( [ 6, 4, 5, 5, 4, 6 ] ),
Transformation( [ 4, 6, 5, 5, 6, 4 ] ),
Transformation( [ 5, 6, 4, 4, 6, 5 ] ),
Transformation( [ 5, 4, 6, 6, 4, 5 ] ) ]
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> IdempotentNC( ker, img ) | ( function ) |
> Idempotent( ker, img ) | ( function ) |
IdempotentNC returns an idempotent with kernel ker and image img without checking IsTransversal (2.2-1) with arguments ker and im.
Idempotent returns an idempotent with kernel ker and image img after checking that IsTransversal (2.2-1) with arguments ker and im returns true.
gap> g1:=Transformation([2,2,4,4,5,6]);; gap> g2:=Transformation([5,3,4,4,6,6]);; gap> ker:=KernelOfTransformation(g2*g1);; gap> im:=ImageListOfTransformation(g2);; gap> Idempotent(ker, im); Error, the image must be a transversal of the kernel [ ... ] gap> Idempotent([[1,2,3],[4,5],[6,7]], [1,5,6]); Transformation( [ 1, 1, 1, 5, 5, 6, 6 ] ) gap> IdempotentNC([[1,2,3],[4,5],[6,7]], [1,5,6]); Transformation( [ 1, 1, 1, 5, 5, 6, 6 ] ) |
> RandomIdempotent( arg ) | ( operation ) |
> RandomIdempotentNC( arg ) | ( operation ) |
If the argument is a kernel, then a random idempotent is return that has that kernel. A kernel is a set of sets that partition the set 1,...n for some n and an image is a sublist of 1,...n.
If the first argument is an image img and the second a positive integer n, then a random idempotent of degree n is returned with image img.
The no check version does not check that the arguments can be the kernel and image of an idempotent.
gap> RandomIdempotent([[1,2,3], [4,5], [6,7,8]], [1,2,3]);; fail gap> RandomIdempotent([1,2,3],5); Transformation( [ 1, 2, 3, 1, 3 ] ) gap> RandomIdempotent([[1,6], [2,4], [3,5]]); Transformation( [ 1, 2, 5, 2, 5, 1 ] ) |
> RandomTransformation( arg ) | ( operation ) |
> RandomTransformationNC( arg ) | ( operation ) |
These are new methods for the existing library function RandomTransformation.
If the first argument is a kernel and the second an image, then a random transformation is returned with this kernel and image.A kernel is a set of sets that partition the set 1,...n for some n and an image is a sublist of 1,...n.
If the argument is a kernel, then a random transformation is returned that has that kernel.
If the first argument is an image img and the second a positive integer n, then a random transformation of degree n is returned with image img.
The no check version does not check that the arguments can be the kernel and image of a transformation.
gap> RandomTransformation([[1,2,3], [4,5], [6,7,8]], [1,2,3]);; Transformation( [ 2, 2, 2, 1, 1, 3, 3, 3 ] ) gap> RandomTransformation([[1,2,3],[5,7],[4,6]]); Transformation( [ 3, 3, 3, 6, 1, 6, 1 ] ) gap> RandomTransformation([[1,2,3],[5,7],[4,6]]); Transformation( [ 4, 4, 4, 7, 3, 7, 3 ] ) gap> RandomTransformationNC([[1,2,3],[5,7],[4,6]]); Transformation( [ 1, 1, 1, 7, 5, 7, 5 ] ) gap> RandomTransformation([1,2,3], 6); Transformation( [ 2, 1, 2, 1, 1, 2 ] ) gap> RandomTransformationNC([1,2,3], 6); Transformation( [ 3, 1, 2, 2, 1, 2 ] ) |
> TransformationActionNC( list, act, elm ) | ( operation ) |
returns the list list acted on by elm via the action act.
gap> mat:=OneMutable(GeneratorsOfGroup(GL(3,3))[1]);
[ [ Z(3)^0, 0*Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0, 0*Z(3) ],
[ 0*Z(3), 0*Z(3), Z(3)^0 ] ]
gap> mat[3][3]:=Z(3)*0;
0*Z(3)
gap> F:=BaseDomain(mat);
GF(3)
gap> TransformationActionNC(Elements(F^3), OnRight, mat);
Transformation( [ 1, 1, 1, 4, 4, 4, 7, 7, 7, 10, 10, 10, 13, 13, 13, 16, 16,
16, 19, 19, 19, 22, 22, 22, 25, 25, 25 ] )
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> IsTransversal( list1, list2 ) | ( function ) |
returns true if the list list2 is a transversal of the list of lists list1. That is, if every list in list1 contains exactly one element in list2.
gap> g1:=Transformation([2,2,4,4,5,6]);; gap> g2:=Transformation([5,3,4,4,6,6]);; gap> ker:=KernelOfTransformation(g2*g1); [ [ 1 ], [ 2, 3, 4 ], [ 5, 6 ] ] gap> im:=ImageListOfTransformation(g2); [ 5, 3, 4, 4, 6, 6 ] gap> IsTransversal(ker, im); false gap> IsTransversal([[1,2,3],[4,5],[6,7]], [1,5,6]); true |
> IsKerImgOfTransformation( ker, img ) | ( function ) |
returns true if the arguments ker and img can be the kernel and image of a single transformation, respectively. The argument ker should be a set of sets that partition the set 1,...n for some n and img should be a sublist of 1,...n.
gap> ker:=[[1,2,3],[5,6],[8]]; [ [ 1, 2, 3 ], [ 5, 6 ], [ 8 ] ] gap> img:=[1,2,9]; [ 1, 2, 9 ] gap> IsKerImgOfTransformation(ker,img); false gap> ker:=[[1,2,3,4],[5,6,7],[8]]; [ [ 1, 2, 3, 4 ], [ 5, 6, 7 ], [ 8 ] ] gap> IsKerImgOfTransformation(ker,img); false gap> img:=[1,2,8]; [ 1, 2, 8 ] gap> IsKerImgOfTransformation(ker,img); true |
> KerImgOfTransformation( f ) | ( operation ) |
returns the kernel and image set of the transformation f. These attributes of f can be obtain separately using KernelOfTransformation (Reference: KernelOfTransformation) and ImageSetOfTransformation (Reference: ImageSetOfTransformation), respectively.
gap> t:=Transformation( [ 10, 8, 7, 2, 8, 2, 2, 6, 4, 1 ] );;
gap> KerImgOfTransformation(t);
[ [ [ 1 ], [ 2, 5 ], [ 3 ], [ 4, 6, 7 ], [ 8 ], [ 9 ], [ 10 ] ],
[ 1, 2, 4, 6, 7, 8, 10 ] ]
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> IsRegularTransformation( S, f ) | ( operation ) |
if f is a regular element of the transformation semigroup S, then true is returned. Otherwise false is returned.
A transformation f is regular inside a transformation semigroup S if it lies inside a regular D-class. This is equivalent to the orbit of the image of f containing a transversal of the kernel of f.
gap> g1:=Transformation([2,2,4,4,5,6]);; gap> g2:=Transformation([5,3,4,4,6,6]);; gap> m1:=Monoid(g1,g2);; gap> IsRegularTransformation(m1, g1); true gap> img:=ImageSetOfTransformation(g1); [ 2, 4, 5, 6 ] gap> ker:=KernelOfTransformation(g1); [ [ 1, 2 ], [ 3, 4 ], [ 5 ], [ 6 ] ] gap> ForAny(MonoidOrbit(m1, img), x-> IsTransversal(ker, x)); true gap> IsRegularTransformation(m1, g2); false gap> IsRegularTransformation(FullTransformationSemigroup(6), g2); true |
> IndexPeriodOfTransformation( f ) | ( attribute ) |
returns the minimum numbers m, r such that f^(m+r)=f^m; known as the index and period of the transformation.
gap> f:=Transformation( [ 3, 4, 4, 6, 1, 3, 3, 7, 1 ] );; gap> IndexPeriodOfTransformation(f); [ 2, 3 ] gap> f^2=f^5; true |
> SmallestIdempotentPower( f ) | ( attribute ) |
returns the least natural number n such that the transformation f^n is an idempotent.
gap> t:=Transformation( [ 6, 7, 4, 1, 7, 4, 6, 1, 3, 4 ] );; gap> SmallestIdempotentPower(t); 6 gap> t:=Transformation( [ 6, 6, 6, 2, 7, 1, 5, 3, 10, 6 ] );; gap> SmallestIdempotentPower(t); 4 |
> InversesOfTransformation( S, f ) | ( operation ) |
> InversesOfTransformationNC( S, f ) | ( operation ) |
returns a list of the inverses of the transformation f in the transformation semigroup S. The function InversesOfTransformationNC does not check that f is an element of S.
gap> S:=Semigroup([ Transformation( [ 3, 1, 4, 2, 5, 2, 1, 6, 1 ] ),
Transformation( [ 5, 7, 8, 8, 7, 5, 9, 1, 9 ] ),
Transformation( [ 7, 6, 2, 8, 4, 7, 5, 8, 3 ] ) ]);;
gap> f:=Transformation( [ 3, 1, 4, 2, 5, 2, 1, 6, 1 ] );;
gap> InversesOfTransformationNC(S, f);
[ ]
gap> IsRegularTransformation(S, f);
false
gap> f:=Transformation( [ 1, 9, 7, 5, 5, 1, 9, 5, 1 ] );;
gap> inv:=InversesOfTransformation(S, f);
[ Transformation( [ 1, 5, 1, 1, 5, 1, 3, 1, 2 ] ),
Transformation( [ 1, 5, 1, 2, 5, 1, 3, 2, 2 ] ),
Transformation( [ 1, 2, 3, 5, 5, 1, 3, 5, 2 ] ) ]
gap> IsRegularTransformation(S, f);
true
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> AsBooleanMatrix( f[, n] ) | ( operation ) |
returns the transformation or permutation f represented as an n by n Boolean matrix where i,f(i)th entries equal 1 and all other entries are 0.
If f is a transformation, then n is the size of the domain of f.
If f is a permutation, then n is the number of points moved by f.
gap> t:=Transformation( [ 4, 2, 2, 1 ] );;
gap> AsBooleanMatrix(t);
[ [ 0, 0, 0, 1 ], [ 0, 1, 0, 0 ], [ 0, 1, 0, 0 ], [ 1, 0, 0, 0 ] ]
gap> t:=(1,4,5);;
gap> AsBooleanMatrix(t);
[ [ 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 0 ] ]
gap> AsBooleanMatrix(t,3);
fail
gap> AsBooleanMatrix(t,5);
[ [ 0, 0, 0, 1, 0 ], [ 0, 1, 0, 0, 0 ], [ 0, 0, 1, 0, 0 ], [ 0, 0, 0, 0, 1 ],
[ 1, 0, 0, 0, 0 ] ]
gap> AsBooleanMatrix(t,6);
[ [ 0, 0, 0, 1, 0, 0 ], [ 0, 1, 0, 0, 0, 0 ], [ 0, 0, 1, 0, 0, 0 ],
[ 0, 0, 0, 0, 1, 0 ], [ 1, 0, 0, 0, 0, 0 ], [ 0, 0, 0, 0, 0, 1 ] ]
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> AsPermOfRange( x ) | ( operation ) |
converts a transformation x that is a permutation of its image into that permutation.
gap> t:=Transformation([1,2,9,9,9,8,8,8,4]); Transformation( [ 1, 2, 9, 9, 9, 8, 8, 8, 4 ] ) gap> AsPermOfRange(t); (4,9) gap> t*last; Transformation( [ 1, 2, 4, 4, 4, 8, 8, 8, 9 ] ) gap> AsPermOfRange(last); () |
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