tactical decomposition

Let ${\cal I}$ be an incidence structure with point set ${\cal P}$ and block set ${\cal B}$ . Let $X_{\cal P}$ be a partition of ${\cal P}$ into classes ${\cal P}_{i}$ , and $X_{\cal B}$ a partition of ${\cal B}$ into classes ${\cal B}_{j}$ . Let $\#(\hbox{\sc p},{\cal B}_{j})$ denote for a moment the number of blocks in class ${\cal B}_{j}$ incident with point p, and $\#(\hbox{\sc b},{\cal P}_{i})$ the number of points in class ${\cal P}_{i}$ incident with block b. Now the pair $(X_{\cal P},X_{\cal B})$ is said to be

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point-tactical iff $\#(\hbox{\sc p},{\cal B}_{j})$ is for any p the same for all ${\cal B}_{j}$ , and is the same for all p within a class ${\cal P}_{i}$ ,
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block-tactical iff $\#(\hbox{\sc b},{\cal P}_{i})$ is for any b the same for all ${\cal P}_{i}$ , and is the same for all b within a class ${\cal B}_{j}$ ,
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a tactical decomposition if both hold.

An incidence structure admitting a tactical decomposition with a single point class ${\cal P}_{0}={\cal P}$ is called resolvable and $X_{\cal B}$ its resolution. Note $\#(\hbox{\sc p},{\cal B}_{j})$ is now a constant throughout. If the constant is 1 the resolution is called a parallelism.

Example of point- and block-tactical: let ${\cal I}$ be simple (it’s a hypergraph) and let $(X_{\cal P},X_{\cal B})$ partition ${\cal P}$ and ${\cal B}$ 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 group $G$ of ${\cal I}$ . It induces a tactical decomposition with as point classes the orbits of $G$ acting on ${\cal P}$ and as block classes the orbits of $G$ acting on ${\cal B}$ .

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