[SONATA-tutorial] 3 The nearring library

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# 3 The nearring library

### Sections

1. An application of the nearring library
2. Appendix K revisited

There are many non-isomorphic nearrings, even of small order. All non-isomorphic nearrings of orders 2 to 15 and all non-isomorphic nearrings with identity up to order 31 with exception of those on the elementary abelian groups of orders 16 and 27 are collected in the SONATA nearring library.

## 3.1 An application of the nearring library

The number of nearrings in the library is big. For example, try

        gap> NumberLibraryNearRings( GTW12_3 );
        48137

Try your favorite small groups with this function to get an impression of these numbers.

Of course, no one can know all these nearrings personally. Therefore, the main purpose of the nearring library is to filter out the nearrings of interest.

Consider for example the following

Problem: How many non-rings with identity of order 4 are there and what do they look like? If you cannot answer this question adhoc, stay tuned.

Let's start with the groups of order 4. Of course you know, there are 2 groups of order 4: `GTW4_1`, the cyclic group and `GTW4_2`, Klein's four group.

Let's go for `GTW4_1` first:

        gap> NumberLibraryNearRingsWithOne( GTW4_1 );
        1
        gap> Filtered( AllLibraryNearRingsWithOne( GTW4_1 ),
        >              n -> not IsDistributiveNearRing( n ) );
        [  ]

So, the only nearring with identity there is on `GTW4_1` is the ring. Well... you knew that before, didn't you?

Now for `GTW4_2`:

        gap> NumberLibraryNearRingsWithOne( GTW4_2 );
        5
        gap> Filtered( AllLibraryNearRingsWithOne( GTW4_2 ),
        >              n -> not IsDistributiveNearRing( n ) );
        [ LibraryNearRing(4/2, 12), LibraryNearRing(4/2, 22) ]

Here we go:

        gap> PrintTable( LibraryNearRing( GTW4_2, 12 ) );
        Let:
        n0 := (())
        n1 := ((3,4))
        n2 := ((1,2))
        n3 := ((1,2)(3,4))

           +  | 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  n0  n2  n2
          n3  | n0  n1  n2  n3
        gap> PrintTable( LibraryNearRing( GTW4_2, 22 ) );
        Let:
        n0 := (())
        n1 := ((3,4))
        n2 := ((1,2))
        n3 := ((1,2)(3,4))

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

## 3.2 Appendix K revisited

An alternative to filtering the nearring library is to use a `for ... do ... od` construction.

We shall demonstrate this by recomputing the list of nearrings given in appendix K of Pilz:Nearrings, i.e. a list of all nearrings on the dihedral group of order 8 (`GTW8_4`) which have an identity, are non-zerosymmetric or are integral.

First, we initialize the variable `nr_list` as the empty list:

        gap> nr_list := [ ];
        [ ]

Now, we write ourselves a `for` loop and add those nearrings we want:

        gap> for i in [1..NumberLibraryNearRings( GTW8_4 )] do
        >       n := LibraryNearRing( GTW8_4, i );
        >       if ( not IsZeroSymmetricNearRing( n ) or
        >            IsIntegralNearRing( n ) or
        >            Identity( n ) <> fail
        >          ) then
        >         Add( nr_list, n );
        >       fi;
        >    od;
        gap> Length( nr_list );
        141

How many boolean nearrings are amongst these? We call a nearring boolean if x*x=x for all x ∈ N.

        gap> Filtered( nr_list, IsBooleanNearRing );
        [ LibraryNearRing(8/4, 1314), LibraryNearRing(8/4, 1380),
          LibraryNearRing(8/4, 1446), LibraryNearRing(8/4, 1447) ]

Which correspond to the numbers 140, 86, 99, and 141 in Pilz:Nearrings, appendix K, accordingly.

For those who got interested in boolean nearrings: many results about them have been collected in Pilz:Nearrings, 9.31.

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