[GAP Forum] Ring by tables

[Junior Assis]

Thank you very much, Peter.
It was helpful.

Now I have another question: Can I construct a structure like I define the
nearrings by tables, but using semigroups instead of groups??

Thanks again.
Junior Assis.



On Wed, Oct 7, 2009 at 9:24 AM, Peter Mayr <peter.mayr at jku.at> wrote:

> Dear Junior Assis
>
> > I would like to know how can I construct a ring by operation tables.
>
> Constructing a ring from tables is a bit awkward but you can use the
> package SONATA for it.
> Let's say we want to build the field of size 2. First construct the
> additive group as a magma from the operation table A (Here i+j :=
> A[i][j]):
>
> gap> A := [[1,2],[2,1]];;
> gap> G := MagmaWithInversesByMultiplicationTable( A );
> <magma-with-inverses with 2 generators>
> gap> IsGroup( G );
> true
>
> Then we attach the multiplicative structure to obtain a near-ring. For
> this you need to load SONATA.
>
> gap> RequirePackage( "sonata" );
>
> We build a binary multiplication function mult on G from the
> multiplication table M. Note that the order of the elements of G in
> elmlist has to correspond to the order of rows and columns in the
> tables A and in M.
>
> gap> M := [[1,1],[1,2]];;
> gap> elmlist := List( [1..Size(G)], i -> MagmaElement( G, i ) );
> [ m1, m2 ]
> gap> mult := NearRingMultiplicationByOperationTable( G, M, elmlist );
> function( x, y ) ... end
>
> Finally we construct a near-ring R whose additive group is isomorphic
> to G and whose multiplication is given by mult.
>
> gap> R := ExplicitMultiplicationNearRing( G, mult );
> ExplicitMultiplicationNearRing ( <group of size 2 with
> 2 generators> , multiplication )
>
> R forms the field of size 2. However, because of its construction R is
> not in the GAP-category of rings but in that of near-rings (A left
> near-ring differs from a ring in that addition is not necessarily
> commutative and only the left distributive law is required).
>
> gap> IsRing( R );
> false
> gap> IsDistributiveNearRing( R );
> true
>
> Consequently not all GAP-functions for rings can be applied to R.
> Please see the SONATA-manual for the methods that are available for
> near-rings.
>
> Hope this helps,
> Peter
>
> --
> Peter Mayr
> CAUL
> Lisbon, Portugal
>