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 partitionof ${\\cal B}$ into classes ${\\cal B}_{j}$ . Let $\\#(\\hbox{\\sc p},{\\cal B}_{j})$ denote for a moment the numberof blocks in class ${\\cal B}_{j}$ incident with point p, and $\\#(\\hbox{\\sc b},{\\cal P}_{i})$ thenumber 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 thesame for all ${\\cal B}_{j}$ , and is the same for all p within aclass ${\\cal P}_{i}$ ,
•
block-tactical iff $\\#(\\hbox{\\sc b},{\\cal P}_{i})$ is for any b thesame for all ${\\cal P}_{i}$ , and is the same for all b within aclass ${\\cal B}_{j}$ ,
•
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 theresolution is called a parallelism.

Example of point- and block-tactical: let ${\\cal I}$ be simple (it’s ahypergraph) and let $(X_{\\cal P},X_{\\cal B})$ partition ${\\cal P}$ and ${\\cal B}$ into a single classeach. This is point-tactical for a regular hypergraph, and block-tactical fora 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 automorphismgroup $G$ of ${\\cal I}$ . It induces a tactical decomposition with as point classesthe orbits of $G$ acting on ${\\cal P}$ and as block classes the orbits of $G$ actingon ${\\cal B}$ .

Trivial example of a tactical decomposition: a partition into singletonpoint and block classes.

The term tactical decomposition (taktische Zerlegung in German) wasintroduced by Peter Dembowski.

单词	TacticalDecomposition
释义	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 partitionof ${\\cal B}$ into classes ${\\cal B}_{j}$ . Let $\\#(\\hbox{\\sc p},{\\cal B}_{j})$ denote for a moment the numberof blocks in class ${\\cal B}_{j}$ incident with point p, and $\\#(\\hbox{\\sc b},{\\cal P}_{i})$ thenumber 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 thesame for all ${\\cal B}_{j}$ , and is the same for all p within aclass ${\\cal P}_{i}$ , • block-tactical iff $\\#(\\hbox{\\sc b},{\\cal P}_{i})$ is for any b thesame for all ${\\cal P}_{i}$ , and is the same for all b within aclass ${\\cal B}_{j}$ , • 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 theresolution is called a parallelism. Example of point- and block-tactical: let ${\\cal I}$ be simple (it’s ahypergraph) and let $(X_{\\cal P},X_{\\cal B})$ partition ${\\cal P}$ and ${\\cal B}$ into a single classeach. This is point-tactical for a regular hypergraph, and block-tactical fora 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 automorphismgroup $G$ of ${\\cal I}$ . It induces a tactical decomposition with as point classesthe orbits of $G$ acting on ${\\cal P}$ and as block classes the orbits of $G$ actingon ${\\cal B}$ . Trivial example of a tactical decomposition: a partition into singletonpoint and block classes. The term tactical decomposition (taktische Zerlegung in German) wasintroduced by Peter Dembowski.
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