Let be an incidence structure with point set and block set . Let be a partition of into classes , and a partition of into classes . Let denote for a moment the number of blocks in class incident with point p, and the number of points in class incident with block b. Now the pair is said to be
point-tactical iff is for any p the same for all , and is the same for all p within a class ,
block-tactical iff is for any b the same for all , and is the same for all b within a class ,
a tactical decomposition if both hold.
An incidence structure admitting a tactical decomposition with a single point class is called resolvable and its resolution. Note is now a constant throughout. If the constant is 1 the resolution is called a parallelism.
Example of point- and block-tactical: let be simple (it’s a hypergraph) and let partition and into a single class each. This is point-tactical for a regular hypergraph, and block-tactical for a uniform hypergraph.
A natural example of a tactical decomposition is provided by the automorphism group of . It induces a tactical decomposition with as point classes the orbits of acting on and as block classes the orbits of acting on .
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.
|Date of creation||2013-03-22 15:11:02|
|Last modified on||2013-03-22 15:11:02|
|Last modified by||marijke (8873)|