[SONATA-tutorial] 2 Nearrings

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# 2 Nearrings

### Sections

1. Entering nearrings into the system
2. Some simple questions about the nearring
3. Entering the nearring with less typing

A (left) nearring is an algebra (N,+,*), where (N,+) is a (not necessarily abelian) group, (N,*) is a semigroup, and the distributive law x*(y+z) = x*y+x*z holds. Such nearrings are called left nearrings. A typical example is constructed as follows: take a group (G,+) (not necessarily abelian), and take the set M(G) of all mappings from G to G. Then we define + on M(G) as pointwise addition of mappings, and * by m * n (γ) : = n (m (γ)). The multiplication looks more natural if we write functions right of their arguments. Then the definition reads (γ) m * n = ((γ)m)n.

Textbooks on nearrings are meldrum85:NATLWG, Clay:Nearrings, Ferrero:Nearrings. They all use left nearrings. The book Pilz:Nearrings uses right nearrings; these are the algebras that arise if we claim the right distributive law (x + y) * z = x*z + y*z instead of the left distributive law given above.

SONATA uses left nearrings throughout.

## 2.1 Entering nearrings into the system

The problem: Input the nearring given in the example of page 406 of Pilz:Nearrings into SONATA.

This nearring is given by an explicit multiplication table. The function `ExplicitMultiplicationNearRing` can be used to do the job. But first, let's get the additive group, which is Klein's four group:

        gap> G := GTW4_2;
        4/2

Now we have to establish a correspondence between the elements `0`, `a`, `b`, `c` of the group in the example and GAP's representation of the group elements.

        gap> AsSortedList( G );
        [ (), (3,4), (1,2), (1,2)(3,4) ]

Ok, let's map `0` to `()`, `a` to `(3,4)`, `b` to `(1,2)` and `c` to `(1,2)(3,4)`

        gap> SetSymbols( G, [ "0", "a", "b", "c" ] );
        gap> PrintTable( G );
        Let:
        0 := ()
        a := (3,4)
        b := (1,2)
        c := (1,2)(3,4)

          +  | 0 a b c
          ------------
          0  | 0 a b c
          a  | a 0 c b
          b  | b c 0 a
          c  | c b a 0

Now for entering the nearring multiplication: We will use the function `NrMultiplicationByOperationTable`. This function requires as one of its arguments a matrix of integers representing the operation table: We choose the entries of `table` according to the positions of the elements of `G` in `AsSortedList( G )`:

        gap> table := [ [ 1, 1, 1, 1 ],
        >               [ 1, 1, 2, 2 ],
        >               [ 1, 2, 4, 3 ],
        >               [ 1, 2, 3, 4 ] ];
        [ [ 1, 1, 1, 1 ], [ 1, 1, 2, 2 ], [ 1, 2, 4, 3 ], [ 1, 2, 3, 4 ] ]

Now we are in position to define a nearring multiplication:

        gap> mul:=NearRingMultiplicationByOperationTable(
        >             G, table, AsSortedList(G) );
        function( x, y ) ... end

And finally, we can define the nearring:

        gap> N := ExplicitMultiplicationNearRing( G, mul );
        ExplicitMultiplicationNearRing ( 4/2 , multiplication )

We get no error message, which means that we have indeed defined a nearring multiplication on `G`. Now let's take a look at it:

        gap> PrintTable( N );
        Let:
        0 := (())
        a := ((3,4))
        b := ((1,2))
        c := ((1,2)(3,4))

          +  | 0  a  b  c
          ---------------
          0  | 0  a  b  c
          a  | a  0  c  b
          b  | b  c  0  a
          c  | c  b  a  0

          *  | 0  a  b  c
          ---------------
          0  | 0  0  0  0
          a  | 0  0  a  a
          b  | 0  a  c  b
          c  | 0  a  b  c

The symbols used for the elements of the group are also used for the elements of the nearring. Of course, it is still possible to redefine the symbols.

## 2.2 Some simple questions about the nearring

Now, that the nearring is in the system, let's ask some questions about it. A nearring is a nearfield if it has more than one element and its nonzero elements are a group with respect to multiplication. A textbook on nearfields is Waehling:Fastkoerper. They are interesting structures, closely connected to sharply 2-transitive permutation groups and fixedpointfree automorphism groups of groups.

        gap> IsNearField( N );
        false
        gap> IsIntegralNearRing( N );
        false
        gap> IsNilpotentNearRing( N );
        false

Pilz:Nearrings is correct ... Well at least in this case.`;-))`

## 2.3 Entering the nearring with less typing

Certainly, everybody has immediately seen, that this nearring is a transformation nearring on `GTW4_2` which is generated by the transformations `0` to `0`, `a` to `a`, `b` to `c`, `c` to `b`, and the identity transformation, so

        gap> t := GroupGeneralMappingByImages(
        >           G, G, AsSortedList(G), AsSortedList(G){[1,2,4,3]} );
        [ (), (3,4), (1,2), (1,2)(3,4) ] -> [ (), (3,4), (1,2)(3,4), (1,2) ]
        gap> id := IdentityMapping( G );
        IdentityMapping( 4/2 )
        gap> T := TransformationNearRingByGenerators( G, [t,id] );
        TransformationNearRingByGenerators(
        [ [ (), (3,4), (1,2), (1,2)(3,4) ] -> [ (), (3,4), (1,2)(3,4), (1,2) ],
          IdentityMapping( 4/2 ) ])

Let's see what we've got:

        gap> PrintTable(T);
        Let:
        n0 := <mapping: 4/2 -> 4/2 >
        n1 := <mapping: 4/2 -> 4/2 >
        n2 := <mapping: 4/2 -> 4/2 >
        n3 := <mapping: 4/2 -> 4/2 >

           +  | 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  n0  n1  n1
          n2  | n0  n1  n2  n3
          n3  | n0  n1  n3  n2

Obviously, we've got the correct nearring. Let's make for sure:

        gap> IsIsomorphicNearRing( N, T );
        true

However, `N` and `T` are certaily not equal:

        gap> N = T;
        false

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