PlanetMath: tactical decomposition

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tactical decomposition

(Definition)

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 $\char93 (\hbox{\sc p},{\cal B}_j)$ denote for a moment the number of blocks in class ${\cal B}_j$ incident with point $\hbox{\sc p}$ , and $\char93 (\hbox{\sc b},{\cal P}_i)$ the number of points in class ${\cal P}_i$ incident with block $\hbox{\sc b}$ . Now the pair $(X_{\cal P},X_{\cal B})$ is said to be

point-tactical iff $\char93 (\hbox{\sc p},{\cal B}_j)$ is for any $\hbox{\sc p}$ the same for all ${\cal B}_j$ , and is the same for all $\hbox{\sc p}$ within a class ${\cal P}_i$ ,
block-tactical iff $\char93 (\hbox{\sc b},{\cal P}_i)$ is for any $\hbox{\sc b}$ the same for all ${\cal P}_i$ , and is the same for all $\hbox{\sc b}$ within a class ${\cal B}_j$ ,
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 $\char93 (\hbox{\sc p},{\cal B}_j)$ 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 of ${\cal I}$ . It induces a tactical decomposition with as point classes the orbits of acting on ${\cal P}$ and as block classes the orbits of 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.

"tactical decomposition" is owned by marijke.

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