# 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

• 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 ${\cal P}_{i}$,

• 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 ${\cal B}_{j}$,

• 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 .

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 TacticalDecomposition 2013-03-22 15:11:02 2013-03-22 15:11:02 marijke (8873) marijke (8873) 5 marijke (8873) Definition msc 05B25 IncidenceStructures point-tactical block-tactical