Plans on incidence structures in general.
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Detailed Description
Plans on incidence structures in general.
 Todo:
 Basic notions

In ComputerAlgebra/Sets we should have the notion of a "correspondence" (following Bourbaki), a triple [R,A,B], where A,B are sets and R is a subset of A x B.

An "incidence structure" is now the version, where the set R is given implicitly, but with values 0,1, that is, an incidence structure is a triple [I,P,B], where P is the set of "points", B the set of "blocks", and I is a 0,1valued function I(p,b).

Abbreviation: "ics".

So [I,P,B] is an incidence structure iff [P,B,I] is a combinatorial matrix over {0,1}; this is useful, however the concepts are different (so for example we have different morphisms).

Natural representations are given by ics2ghg and ghg2ics.

We should also introduce the notion of a "polar incidence structure", a pair [I,S] s.t. [I,S,S] is an incidence structure, and I is symmetric.

These are the structures which correspond to incidence structures together with a *given* polarity.

Such polar incidence structures correspond 11 to graphs with loops (I is just the 01representation of the the adjacencyrelation), and they also have the same morphisms (so we have a nonfull subcategory of the category of all incidence structures).

See "Graphs as hypergraphs" in ComputerAlgebra/Graphs/Lisp/plans/general.hpp, and also ComputerAlgebra/Graphs/Lisp/StrongRegularity/plans/general.hpp.

Abbreviation: "pics".

Natural representations are given by pics2pghg and pghg2pics, and by pics2gl and gl2pics.
 Todo:
 The main parameters

Apparently there is no useful notion for the main parameters, that "for t different points there are exactly lambda incident blocks".

So we need to introduce it ourselves: An incidence structure [I,P,B] has *index lambda in NN_0 w.r.t. t points (t in NN_0)* if for each subset T of P of size t the subset of B of blocks incident with all elements of T has size lambda.

The predicate is "balanced_ics_p(S,t,lambda)".
 Todo:
 Polar and bipolar incidence structures

A "bipolar incidence structure" is a pair [I,V] such that [I,V,V] is an incidence structure.

A "polar incidence structure" is a bipolar incidence structure [I,V] such that I is symmetric.
Definition in file general.hpp.