5.7 ActionWithZero

ActionWithZero( M, D )
ActionWithZero( M, D, action )

ActionWithZero returns a transformation monoid with the same number of generators as M, such that each generator of the transformation monoid acts on the set [1..Length(D)+1] in the same way as the corresponding generator of the monoid M acts on the domain <D> cup {0}, which may be a list of arbitrary type.

Here it is not required that D be invariant under M. Whenever the image of a point d under the monoid element m does not lie in D it is set to 0. The image of 0 under every monoid element is set to 0. Note that this way the resulting monoid is a homomorphic image of M if and only if D is a union of strong orbits. The point 0 is represented by Length(D) + 1 in the domain of the transformation monoid returned by ActionWithZero.

ActionWithZero accepts a function action of two arguments d and m as optional third argument, which specifies how the elements of M act on D (see Other Actions).

ActionWithZero calls
M.operations.ActionWithZero( M, D, action )
and returns the value. Note that the third argument is not optional for functions called this way.

The default function called this way is MonoidOps.ActionWithZero, which simply applies each generator of M to all the points of D, finds the position of the image in D, and finally constructs the transformation (see Transformation) defined by the list of those positions and Length(D)+1 for every image not in D.

    gap> M:= Monoid( [ Transformation( [ 5, 4, 4, 2, 2 ] ), 
      Transformation( [ 2, 5, 5, 4, 1 ] ) ] );;
    gap> M.name:= "M";;
    gap> class:= LClass( M, Transformation( [ 1, 4, 4, 5, 5 ] ) );
    LClass( M, Transformation( [ 1, 4, 4, 5, 5 ] ) )
    gap> orb:= ShortOrbit(M, class, OnLClasses, Rank);
    [ LClass( M, Transformation( [ 1, 4, 4, 5, 5 ] ) ), 
      LClass( M, Transformation( [ 2, 4, 4, 1, 1 ] ) ), 
      LClass( M, Transformation( [ 4, 2, 2, 5, 5 ] ) ) ]
    gap> ActionWithZero(M, orb, OnLClasses);
    Monoid( [ Transformation( [ 4, 3, 4, 4 ] ), 
     Transformation( [ 2, 3, 1, 4 ] ) ] )

  
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