% \Chapter{Weakly divisible nearrings} % Weakly divisible nearrings are currently investigated in Brescia and provide a new class of designs different from planar nearrings. A right nearring $(N,+,\cdot)$ is called *weakly divisible* if $\forall a,b\in N \exists x\in N : x\cdot a=b$ or $x\cdot b=a$. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Section{Construction of weakly divisible nearrings on cyclic groups} *The problem:* Find a wd-nearring of order 16. *The solution:* In their paper wd-nearrings on the group of integers (mod $p^n$ ) Anna Benini and Fiorenza Morini presented a method how to construct all wd nearrings with maximal nilpotent ideal on cyclic groups of prime power order. We simply follow their ideas starting with the cyclic group of order 16. Unfortunately `G' is no group of integers but presented as a pc- group. Nevertheless a generator of `G' corresponds to `1' in $\Z_{16}$ and the $i$-th power of the generator corresponds to `i' in $\Z_{16}$. \beginexample gap> G := CyclicGroup( 16 ); <pc group of size 16 with 4 generators> \endexample Do not worry about `G' having 4 generators; the first one generates the whole group. \beginexample gap> g := GeneratorsOfGroup( G )[1]; f1 gap> Order( g ); 16 \endexample What we need next for cooking up our wd-nearring is some automorphism group on `G'. We feel free to choose whatever we like. Let `a7' be the multiplication of each group element by 7, `a9' the multiplication by 9, and `phi' the automorphism group generated by both. `GroupHomomorphismByImages( <G>, <G>, <gens>, <imgs> )' returns the homomorphism from <G> to <G> that takes each generator in the list <gens> to its image in the list <imgs>. \beginexample gap> a7 := GroupHomomorphismByImages( G, G, [g], [g^7] ); [ f1 ] -> [ f1*f2*f3 ] gap> a9 := GroupHomomorphismByImages( G, G, [g], [g^9] ); [ f1 ] -> [ f1*f4 ] gap> phi := Group( a7, a9 ); <group with 2 generators> gap> Size( phi ); 4 \endexample All nearring multiplications on a cyclic group can be described via a *Clay function* and so can wd - nearring multiplications. We know how to build a Clay function for a wd- nearring out of `phi', and some orbit representatives of the group elements of $\Z_{p^n}$ that are relatively prime to $p$ but not every set of representatives is permitted. If two orbits are $p$-comparable the representatives have to be chosen congruent mod $p$ resp. $4$ depending on $p$ and on the automorphism group $\Phi$. `PComparableOrbitsRepresentatives( <G>, <phi> )' helps choosing representatives that fulfill all the necessary conditions. The orbits are gathered in sets of $p$-comparable orbits while the elements of each orbit are separated into subsets of elements congruent to each other modulo $p$ resp. $4$. All we have to do find a a permissible set of representatives is to choose in each set of $p$-comparable orbits one element of the $i$-th subset. \beginexample gap> Orbits( phi, G ); [ [ <identity> of ... ], [ f4 ], [ f3, f3*f4 ], [ f2, f2*f3*f4 ], [ f2*f4, f2*f3 ], [ f1, f1*f2*f3, f1*f4, f1*f2*f3*f4 ], [ f1*f3, f1*f2, f1*f3*f4, f1*f2*f4 ] ] gap> pcreps := PComparableOrbitsRepresentatives( phi, G ); [ [ [ [ f1, f1*f4 ], [ f1*f2*f3, f1*f2*f3*f4 ] ], [ [ f1*f3, f1*f3*f4 ], [ f1*f2, f1*f2*f4 ] ] ] ] gap> Length( pcreps ); 1 gap> c := pcreps[1]; [ [ [ f1, f1*f4 ], [ f1*f2*f3, f1*f2*f3*f4 ] ], [ [ f1*f3, f1*f3*f4 ], [ f1*f2, f1*f2*f4 ] ] ] gap> o1 := c[1]; [ [ f1, f1*f4 ], [ f1*f2*f3, f1*f2*f3*f4 ] ] gap> o2 := c[2]; [ [ f1*f3, f1*f3*f4 ], [ f1*f2, f1*f2*f4 ] ] \endexample There are two orbits `o1', `o2' containing 4 group elements relatively prime to 2 each. As there is only one class `c' of $p$-comparable orbits, `Length( pcreps )'$=1$, these two orbits are $p$-comparable. We can either choose one representative out of the first or second subset of each orbit. We don't mind and take them from the second class. Let `e' be some choice element among our representatives in `reps'. \beginexample gap> r1 := o1[2][1]; f1*f2*f3 gap> r2 := o2[2][2]; f1*f2*f4 gap> reps := [ r1, r2 ];; gap> e := r1;; \endexample By now we have all we need to define a Clay function for a wd - nearring on `G'. We got an automorphism group `phi', orbit representatives `reps', which fulfill the necessary conditions, with one distinguished element `e' among them. First we generate the Clay function `pi' out of these and then a wd - `ExplicitMultiplicationNearRing' out of `pi'. \beginexample gap> pi := ClayFunctionForWdNearRing( G, phi, reps, e ); function( a ) ... end gap> nr := NearRingOnCyclicGroupByClayFunction( G, pi ); ExplicitMultiplicationNearRing ( <pc group of size 16 with 4 generators> , multiplication ) \endexample `NearRingOnCyclicGroupByClayFunction' returns a nearring for all kind of Clay - functions on a cyclic group although in general this won't be a wd - nearring. %%% Local Variables: %%% mode: latex %%% TeX-master: "manual" %%% End: