general.hpp File Reference

Plans on incidence structures in general. More...

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Detailed Description

Plans on incidence structures in general.

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 0,1-valued 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.
    1. These are the structures which correspond to incidence structures together with a *given* polarity.
    2. Such polar incidence structures correspond 1-1 to graphs with loops (I is just the 0-1-representation of the the adjacency-relation), and they also have the same morphisms (so we have a non-full subcategory of the category of all incidence structures).
    3. 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.
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)".
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.