62.32 Nearring

Nearring( G, mul )

In this form the constructor function Nearring returns the near-ring defined by the permutation group G and the near-ring multiplication mul. (For a detailed explanation of mul see IsNrMultiplication). Near-ring calls Is-Nr-Mul-ti-pli-ca-tion in order to make sure that mul is really a near-ring multiplication.

  gap> g := Group( (1,2,3) );
  Group( (1,2,3) )
  gap> mul_r := function(x,y) return x; end;
  function ( x, y ) ... end
  gap> n := Nearring( g, mul_r );
  Nearring( Group( (1,2,3) ), function ( x, y )
      return x;
  end )
  gap> DisplayCayleyTable( n );
  Let:
  n0 := ()
  n1 := (1,2,3)
  n2 := (1,3,2)

    +  

 n0 n1 n2 
   ---------------
    n0 

 n0 n1 n2 
    n1 

 n1 n2 n0 
    n2 

 n2 n0 n1 

    *  

 n0 n1 n2 
   ---------------
    n0 

 n0 n0 n0 
    n1 

 n1 n1 n1 
    n2 

 n2 n2 n2 

Nearring( t_1, ..., t_n )
Nearring( [t_1, ..., t_n] )

In this form the constructor function Near-ring returns the near-ring generated by the group transformations t_1,dots,t_n. All of them must be transformations on the same permutation group.

Note that Near-ring allows also a list of group transformations as argument, which makes it possible to call Nearring e.g. with a list of endomorphisms generated by the function Endo-mor-phisms (see Endomorphisms for groups), which for a group G allows to compute E(G); Near-ring called with the list of all inner automorphisms of a group G would return I(G).

  gap> t := Transformation( Group( (1,2) ), [2,1] );
  Transformation( Group( (1,2) ), [ 2, 1 ] )
  gap> n := Nearring( t );
  Nearring( Transformation( Group( (1,2) ), [ 2, 1 ] ) ) 
  gap> DisplayCayleyTable( n );
  Let:
  n0 := Transformation( Group( (1,2) ), [ 1, 1 ] )
  n1 := Transformation( Group( (1,2) ), [ 1, 2 ] )
  n2 := Transformation( Group( (1,2) ), [ 2, 1 ] )
  n3 := Transformation( Group( (1,2) ), [ 2, 2 ] )

    +  

 n0 n1 n2 n3 
   ------------------
    n0 

 n0 n1 n2 n3 
    n1 

 n1 n0 n3 n2 
    n2 

 n2 n3 n0 n1 
    n3 

 n3 n2 n1 n0 

    *  

 n0 n1 n2 n3 
   ------------------
    n0 

 n0 n0 n0 n0 
    n1 

 n0 n1 n2 n3 
    n2 

 n3 n2 n1 n0 
    n3 

 n3 n3 n3 n3 

  gap> g := Group( (1,2), (1,2,3) );
  Group( (1,2), (1,2,3) )
  gap> e := Endomorphisms( g );
  [ Transformation( Group( (1,2), (1,2,3) ), [ 1, 1, 1, 1, 1, 1 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 2, 1, 1, 2 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 6, 5, 4, 3 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 2, 5, 4, 6 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 3, 1, 1, 3 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 6, 4, 5, 2 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 2, 4, 5, 3 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 3, 5, 4, 2 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 6, 1, 1, 6 ] ), 
    Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 3, 4, 5, 6 ] ) ]
  gap> nr := Nearring( e );     # the endomorphisms near-ring on S3
  Nearring( Transformation( Group( (1,2), (1,2,3) ), [ 1, 1, 1, 1, 1, 1 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 2, 1, 1, 2 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 3, 4, 5, 6 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 2, 6, 5, 4, 3 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 2, 5, 4, 6 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 3, 1, 1, 3 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 3, 6, 4, 5, 2 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 2, 4, 5, 3 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 3, 5, 4, 2 
   ] ), Transformation( Group( (1,2), (1,2,3) ), [ 1, 6, 6, 1, 1, 6 ] ) ) 
  gap> Size( nr );
  54

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