62.68 Nearring records

The record of a nearring has the following components:

isDomain, isNearring:

these two are always true for a near-ring. isTransformationNearring:
this is bound and true only for transformation near-rings (i.e. those near-rings that are generated by group transformations by using the constructor function Nearring in the second form).

generators:

this is bound only for a transformation near-ring and holds the set of generators of the transformation near-ring.

group:

this component holds the additive group of the near-ring as permutation group.

addition, subtraction, multiplication:

these record fields contain the functions that represent the binary operations that can be performed with the elements of the near-ring on the additive group of the near-ring (addition, subtraction) resp. on the multiplicative semigroup of the near-ring (multiplication)

  gap> nr := Nearring( Transformation( Group( (1,2) ), [2,1] ) );
  Nearring( Transformation( Group( (1,2) ), [ 2, 1 ] ) ) 
  gap> e := Elements( nr );
  [ Transformation( Group( (1,2) ), [ 1, 1 ] ), 
    Transformation( Group( (1,2) ), [ 1, 2 ] ), 
    Transformation( Group( (1,2) ), [ 2, 1 ] ), 
    Transformation( Group( (1,2) ), [ 2, 2 ] ) ]
  gap> nr. addition( e[2], e[3] );
  Transformation( Group( (1,2) ), [ 2, 2 ] )
  gap> nr.multiplication( e[2], e[4] );
  Transformation( Group( (1,2) ), [ 2, 2 ] )
  gap> nr.multiplication( e[2], e[3] );
  Transformation( Group( (1,2) ), [ 2, 1 ] )

operations:

this is the operations record of the near-ring.

size, elements, endomorphisms,

automorphisms:

these entries become bound if the according functions have been performed on the near-ring.

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