PreImages( map, img )
In the first form PreImages
returns the set of elements from the source
of the mapping map that are mapped by map to the element img in the
range of map, i.e., the set of elements elm such that img in
Images( map, elm )
(see Images). map may be a multi valued
mapping.
gap> g := Group( (1,2,3,4), (2,4), (5,6,7) );; g.name := "g";; gap> p4 := MappingByFunction( g, g, x -> x^4 ); MappingByFunction( g, g, function ( x ) return x ^ 4; end ) gap> PreImages( p4, (5,6,7) ); [ (5,6,7), (2,4)(5,6,7), (1,2)(3,4)(5,6,7), (1,2,3,4)(5,6,7), (1,3)(5,6,7), (1,3)(2,4)(5,6,7), (1,4,3,2)(5,6,7), (1,4)(2,3)(5,6,7) ] gap> p5 := MappingByFunction( g, g, x -> x^5 ); MappingByFunction( g, g, function ( x ) return x ^ 5; end ) gap> PreImages( p5, (2,4)(5,6,7) ); [ (2,4)(5,7,6) ]
PreImages( map, imgs )
In the second form PreImages
returns the set of all preimages of the
elements in the set of elements imgs, i.e., the union of the preimages
of the single elements of imgs. map may be a multi valued mapping.
gap> PreImages( p4, [ (), (5,6,7) ] ); [ (), (5,6,7), (2,4), (2,4)(5,6,7), (1,2)(3,4), (1,2)(3,4)(5,6,7), (1,2,3,4), (1,2,3,4)(5,6,7), (1,3), (1,3)(5,6,7), (1,3)(2,4), (1,3)(2,4)(5,6,7), (1,4,3,2), (1,4,3,2)(5,6,7), (1,4)(2,3), (1,4)(2,3)(5,6,7) ] gap> PreImages( p5, [ (), (5,6,7) ] ); [ (), (5,7,6) ]
PreImages
first checks in which form it is called.
In the first case it calls map.operations.PreImagesElm( map, img
)
and returns this value.
The default function called this way is MappingOps.PreImagesElm
, which
runs through all elements of the source of map, if it is finite, and
returns the set of those that have img in their images. Look in the
index under PreImages to see for which mappings this function is
overlaid.
In the second case if calls map.operations.PreImagesSet( map, imgs
)
and returns this value.
The default function called this way is MappingOps.PreImagesSet
, which
returns the union of the preimages of all the elements of the set imgs.
Look in the index under PreImages to see for which mappings this
function is overlaid.
GAP 3.4.4