62.29 Transformation semigroup records

Transformation Semigroups are implemented as records. Such a transformation semigroup record has the following components:

isDomain, isSemigroup:

these two are always true for a transformation semigroup.

isTransformationSemigroup:

this is bound and true only for transformation semigroups.

generators:

this holds the set of generators of a transformation semigroup.

multiplication:

this record field contains a function that represents the binary operation of the semigroup that can be performed on the elements of the semigroup. For transformation semigroups this equals of course, composition. Example:

  gap> elms := Elements( s );
  [ Transformation( [ 1, 2, 3 ], [ 1, 1, 1 ] ), 
    Transformation( [ 1, 2, 3 ], [ 1, 1, 2 ] ), 
    Transformation( [ 1, 2, 3 ], [ 1, 2, 2 ] ), 
    Transformation( [ 1, 2, 3 ], [ 2, 2, 2 ] ), 
    Transformation( [ 1, 2, 3 ], [ 2, 2, 3 ] ), 
    Transformation( [ 1, 2, 3 ], [ 2, 3, 3 ] ), 
    Transformation( [ 1, 2, 3 ], [ 3, 3, 3 ] ) ]
  gap> s.multiplication( elms[5], elms[2] );
  Transformation( [ 1, 2, 3 ], [ 1, 1, 2 ] )

operations:

this is the operations record of the semigroup.

size, elements, rank, smallestIdeal,

IsFinite, identity:

these entries become bound if the according functions have been performed on the semigroup.

GreenL, GreenR, GreenD,

GreenJ, GreenH:

these are entries according to calls of the function tt Green with the corresponding parameters.

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