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 _______________________________________________ Forum mailing list Forum@mail.gap-system.org http://mail.gap-system.org/mailman/listinfo/forum
