62.78 Defining near-rings with known multiplication table

Suppose that for a given group g the multiplication table of a binary operation * on the elements of g is known such that * is a near-ring multiplication on g. Then the constructor function Nearring can be used to input the near-ring specified by g and *.

An example shall illustrate a possibility how this could be accomplished: Take the group S_3, which in GAP can be defined e.g. by

  gap> g := Group( (1,2), (1,2,3) );
  Group( (1,2), (1,2,3) )

This group has the following list of elements:

  gap> Elements( g );
  [ (), (2,3), (1,2), (1,2,3), (1,3,2), (1,3) ]

Let 1 stand for the first element in this list, 2 for the second, and so on up to 6 which stands for the sixth element in the following multiplication table:

catcode`|=12 begintabularccccccc} \catcode`\=13 * & 1 & 2 & 3 & 4 & 5 & 6
hline 1 & 1 & 1 & 1 & 1 & 1 & 1
2 & 2 & 2 & 2 & 2 & 2 & 2
3 & 2 & 2 & 6 & 3 & 6 & 3
4 & 1 & 1 & 5 & 4 & 5 & 4
5 & 1 & 1 & 4 & 5 & 4 & 5
6 & 2 & 2 & 3 & 6 & 3 & 6 A near-ring on g with this multiplication can be input by first defining a multiplication function, say m in the following way:

  gap> m := function( x, y )         
  >   local elms, table;
  >     elms := Elements( g );
  >     table := [ [ 1, 1, 1, 1, 1, 1 ],
  >                [ 2, 2, 2, 2, 2, 2 ],
  >                [ 2, 2, 6, 3, 6, 3 ],
  >                [ 1, 1, 5, 4, 5, 4 ],
  >                [ 1, 1, 4, 5, 4, 5 ],
  >                [ 2, 2, 3, 6, 3, 6 ] ];
  >    
  >     return elms[ table[Position( elms, x )][Position( elms, y )] ];
  >   end;
  function ( x, y ) ... end

Then the near-ring can be constructed by calling Nearring with the parameters g and m:

  gap> n := Nearring( g, m );
  Nearring( Group( (1,2), (1,2,3) ), function ( x, y )
      local  elms, table;
      elms := Elements( g );
      table := [ [ 1, 1, 1, 1, 1, 1 ], [ 2, 2, 2, 2, 2, 2 ], 
          [ 2, 2, 6, 3, 6, 3 ], [ 1, 1, 5, 4, 5, 4 ], 
          [ 1, 1, 4, 5, 4, 5 ], [ 2, 2, 3, 6, 3, 6 ] ];
      return elms[table[Position( elms, x )][Position( elms, y )]];
  end )