An incidence structure created by Magma consists of three objects: the point-set P, the block-set B and the incidence structure D itself.
Although called the point-set and block-set, P and B are not actual Magma sets. They simply act as the parent structures for the points and blocks (respectively) of the incidence structure D, enabling easy creation of these objects via the ! and . operators.
The point-set P belongs to the Magma category IncPtSet, and the block-set B to the category IncBlkSet.
In this section, the functions
used to create point-sets, block-sets and the points and blocks
themselves are described.
Creating Point-Sets and Block-Sets
As mentioned above, the point-set and block-set are returned as the
second and third arguments of any function which creates an
incidence structure. They can also be created via the following
two functions.
PointSet(D) : Inc -> IncPtSet
Given an incidence structure D, return the point-set P of D.
Given an incidence structure D, return the block-set B of D.
For efficency and clarity, the points and blocks of an incidence
structure are given special types in Magma.
The category names for points and blocks
are IncPt and IncBlk respectively. They can be
created in the following ways.
Point(D, i) : Inc, RngIntElt -> IncPt
The i-th point of the incidence structure D.
Given the point-set P of an incidence structure D and an integer i, return the i-th point of D.
Given the point-set P of an incidence structure D, return a representative point of D.
Given the point-set P of an incidence structure D, return a random point of D.
Given the point-set P of an incidence structure D, return the point of D corresponding to the element x of the indexed set used to create D.
The i-th block of the incidence structure D.
Given the block-set B of an incidence structure D and an integer i, return the i-th block of D.
Given the block-set B of an incidence structure D, return a representative block of D.
Given the block-set B of an incidence structure D, return a random block of D.
Given the block-set B of an incidence structure D, and a set S, tries to coerce S into B.
Given a block b of an incidence structure D, return a representative point of D which is incident with b.
Given a block b of an incidence structure D, return a random point incident with b.
> V := {@ 2, 4, 6, 8, 10 @};
> D, P, B := IncidenceStructure< V | {2, 4, 6}, {2, 8, 10}, {4, 6, 8} >;
> D;
Incidence Structure on 5 points with 3 blocks
> P;
Point-set of Incidence Structure on 5 points with 3 blocks
> B;
Block-set of Incidence Structure on 5 points with 3 blocks
> B.2;
{2, 8, 10}
> P.4;
8
> P!4;
4
> P.5 eq Point(D, 5);
true
> b := Random(B);
> b;
{2, 4, 6}
> Parent(b);
Block-set of Incidence Structure on 5 points with 3 blocks
> p := Rep(b);
> p;
2
> Parent(p);
Point-set of Incidence Structure on 5 points with 3 blocks
> B!{2, 8, 10};
{2, 8, 10}