Let $\I$ be an incidence structure with point set $\P$ and block set $\B$ . Let $X_\P$ be a partition of $\P$ into classes $\P_i$ , and $X_\B$ a partition of $\B$ into classes $\B_j$ . Let $\#(\0p,\B_j)$ denote for a moment the number of blocks in class $\B_j$ incident with point $\0p$ , and $\#(\0b,\P_i)$ the number of points in class $\P_i$ incident with block $\0b$ . Now the pair $(X_\P,X_\B)$ is said to be

• point-tactical iff $\#(\0p,\B_j)$ is for any $\0p$ the same for all $\B_j$ , and is the same for all $\0p$ within a class $\P_i$ ,
• block-tactical iff $\#(\0b,\P_i)$ is for any $\0b$ the same for all $\P_i$ , and is the same for all $\0b$ within a class $\B_j$ ,
• a tactical decomposition if both hold.

An incidence structure admitting a tactical decomposition with a single point class $\P_0=\P$ is called resolvable and $X_\B$ its resolution. Note $\#(\0p,\B_j)$ is now a constant throughout. If the constant is 1 the resolution is called a parallelism.

Example of point- and block-tactical: let $\I$ be simple (it's a hypergraph) and let $(X_\P,X_\B)$ 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 group $G$ 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.