In section Transformations we introduced transformations on sets and groups.
We used set transformations together with composition * to construct
transformation semigroups in section Transformation Semigroups.
In section Transformations
we also introduced the operation of pointwise addition + for group
transformations. Now we are able to use these group transformations
together with pointwise addition + and composition * to
construct (right) near-rings.
A (right) near-ring is a nonempty set N together with two binary operations on N, + and cdot s.t. (N,+) is a group, (N,cdot) is a semigroup, and cdot is right distributive over +, i.e. forall n_1,n_2,n_3 in N: (n_1+n_2)cdot n_3 = n_1cdot n_3+n_2cdot n_3.
Here we have to make a remark: we let transformations act from the right; yet in order to get a right transformation near-ring transformations must act from the left, hence we define a near-ring multiplication cdot of two transformations, t_1, t_2 as t_1 cdot t_2 := t_2 * t_1.
There are three possibilities to get a near-ring in GAP: the
constructor function Nearring can be used in two different ways or a
near-ring can be extracted from the near-rings library by using the function
LibraryNearring. All functions described here were programmed for
permutation groups and they also work fine with them; other types of groups
(such as AG groups) are not supported.
Near-rings are represented by records that contain the necessary information to identify them and to do computations with them.
GAP 3.4.4