tactical decomposition

Let ℐ be an incidence structure with point set 𝒫 and block set ℬ. Let X𝒫 be a partitionMathworldPlanetmath of 𝒫 into classes 𝒫i, and Xℬ a partition of ℬ into classes ℬj. Let #⁒(p,ℬj) denote for a moment the number of blocks in class ℬj incidentMathworldPlanetmathPlanetmath with point p, and #⁒(b,𝒫i) the number of points in class 𝒫i incident with block b. Now the pair (X𝒫,Xℬ) is said to be

  • β€’

    point-tactical iff #⁒(p,ℬj) is for any p the same for all ℬj, and is the same for all p within a class 𝒫i,

  • β€’

    block-tactical iff #⁒(b,𝒫i) is for any b the same for all 𝒫i, and is the same for all b within a class ℬj,

  • β€’

    a tactical decomposition if both hold.

An incidence structure admitting a tactical decomposition with a single point class 𝒫0=𝒫 is called resolvable and Xℬ its resolution. Note #⁒(p,ℬj) is now a constant throughout. If the constant is 1 the resolution is called a parallelismPlanetmathPlanetmath.

Example of point- and block-tactical: let ℐ be simple (it’s a hypergraphMathworldPlanetmath) and let (X𝒫,Xℬ) partition 𝒫 and ℬ into a single class each. This is point-tactical for a regular hypergraph, and block-tactical for a uniform hypergraph.

Example of parallelism: an affine plane (lines are the blocks, with parallel ones in the same class).

A natural example of a tactical decomposition is provided by the automorphism groupMathworldPlanetmath G of ℐ. It induces a tactical decomposition with as point classes the orbits of G acting on 𝒫 and as block classes the orbits of G acting on ℬ.

Trivial example of a tactical decomposition: a partition into singleton point and block classes.

The term tactical decomposition (taktische Zerlegung in German) was introduced by Peter Dembowski.

Title tactical decomposition
Canonical name TacticalDecomposition
Date of creation 2013-03-22 15:11:02
Last modified on 2013-03-22 15:11:02
Owner marijke (8873)
Last modified by marijke (8873)
Numerical id 5
Author marijke (8873)
Entry type Definition
Classification msc 05B25
Related topic IncidenceStructures
Defines point-tactical
Defines block-tactical