05B25-TacticalDecomposition.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{TacticalDecomposition}
\pmcreated{2013-03-22 15:11:02}
\pmmodified{2013-03-22 15:11:02}
\pmowner{marijke}{8873}
\pmmodifier{marijke}{8873}
\pmtitle{tactical decomposition}
\pmrecord{5}{36939}
\pmprivacy{1}
\pmauthor{marijke}{8873}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{05B25}
%\pmkeywords{incidence}
%\pmkeywords{design}
%\pmkeywords{geometry}
\pmrelated{IncidenceStructures}
\pmdefines{point-tactical}
\pmdefines{block-tactical}

\endmetadata

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\begin{document}
\PMlinkescapeword{class}
\PMlinkescapeword{classes}
\PMlinkescapeword{constant}
\PMlinkescapeword{decomposition}
\PMlinkescapeword{incident}
\PMlinkescapeword{moment}
\PMlinkescapeword{simple}
\PMlinkescapeword{term}

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
%
\begin{itemize}

\item {\bf 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 $\P_i$,

\item {\bf 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 $\B_j$,

\item a {\bf tactical decomposition} if both hold.

\end{itemize}
%
An incidence structure admitting a tactical decomposition with a single point class $\P_0=\P$ is called {\bf resolvable} and $X_\B$ its {\bf resolution}. Note
$\#(\0p,\B_j)$ is now a constant throughout. If the constant is 1 the
resolution is called a {\bf 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 ({\em taktische Zerlegung\/} in German) was
introduced by Peter Dembowski.
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\end{document}