Even Set Systems

Andreas Dress^{1, 2}

dress@sibs.ac.cn, dress@mis.mpg.de

Annals of Combinatorics 13 (2) pp.193-198 June, 2009

Abstract:

In phylogenetic
combinatorics, the analysis of split systems is a
fundamental issue. Here, we observe that there is a
canonical one-to-one correspondence between split
systems on the one, and ``even'' set systems on the
other hand, i.e., given any finite set X, we show
that there is a canonical one-to-one correspondence
between the set $\p(\sx)$ consisting of all subsets
$\s$ of the set $\sx$ of all splits of the set X
(that is, all 2-subsets {A,B} of the power set
$\px$ of $X$ for which $A\cup B = X$ and $A\cap B =
\emptyset$ hold) and the set $\p^{even}(\px)$
consisting of all subsets $\e$ of the power set $\px$
of $X$ for which, for each subset $Y$ of $X$, the
number of proper subsets of Y contained in $\e$ is
even.
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