A split can be thought of a division of a set into two disjoint subsets. More precisely, a split =
of a set *X* is a partition of *X* into two, non-empty, disjo=
int sets *A* and *B*, so that the union of *A* and *B is equal to X. For example, { {French, English} , {Chinese=
, Russian, Italian} } is a split of the set {French, English, Chinese, Russ=
ian, Italian}. To ease notation, a split {A,B} of a set X=
is sometimes denoted by A|B (or B|A as the order in whic=
h A and B are listed does not matter). Using this notatio=
n, the split above becomes {French, English} | {Chinese, Russian, Italian}.=
*

A *trivial split* of a set *X* is any split of *X* =
of the form {*x*}| *X*-{*x*}, where *x* is some=
element of *X*. For example, {French} | {Chinese, Russian, Italian}=
is a trivial split of the set {French, Chinese, Russian, Italian}.

A *split system* =E2=88=91 on a set *X* is a set of splits=
of *X*. For example, { {French, English} | {German, Italian}, {Fren=
ch, German} | {English, Italian} }, is a split system on the set {French, E=
nglish, German, Italian}, that contains two splits.

Two distinct splits *A*|*B* and *C*|*D* of a=
set are called *compatible* if one of the intersections *A* =
=E2=88=A9 *C*, *A* =E2=88=A9 *D*, *B* =E2=88=A9=
*C*, and *B* =E2=88=A9 *D* is empty. Two splits are c=
alled *incompatible* if they are not compatible. Note that in a phylogenetic tree with =
leaf set *X*, each of the=
edges induces a split of *X<=
/em>, and that the two splits induced by any pair of distinct edges in the =
tree are compatible.*

A *compatible* split system on a set *X* is a split system=
on *X* in which any pair of splits is compatible. Note that it was =
proved by Peter Buneman (1971) that a compatible split system which contain=
s all possible trivial splits can always be represented by a (unique) phylo=
genetic tree with leaf set *X* (see e.g. Theorem 3.1.4 [2]). In part=
icular, each of the edges in the tree represents a split since its removal =
from the tree cuts the tree into two pieces which gives a split of the leaf=
set *X*. This implies that a split system =E2=88=91 which contains =
all possible trivial splits can be represented by a phylogenetic tree if an=
d only if every pair of splits in =E2=88=91 is compatible.

A *hierarchy* is a set *H* of clusters in a set *X*=
so that for any pair of sets *A,B* in *H*, either *A*=
is a subset of *B*, *B* is a subset of *A*, or *A<=
/em> and B are disjoint. Note that a hierarchy on X that =
contains all trivial clusters can always be represented by a rooted, phylog=
enetic tree (see e.g. Theorem 3.5.2 in Semple & Steel 2003).*

A *split network* on a set *X* is a special type of graph =
in which some subset of the vertices are labeled by elements of *X*,=
and certain collections of edges of the graph represent splits of *X, just as the edges in a phylogenetic tree represent splits (see e.g. Sec=
tion 4.3 in Huson et al. 2010). Split networks are often used to represent =
split systems which contain some pairs of incompatible splits, since such s=
plit systems cannot be represented in any phylogenetic tree (see a=
bove "compatible split systems").*

Cf. also the entry set.

*=E2=80=93 Buneman, Peter. 1971. =E2=80=9CThe Recovery of Trees from Meas=
ures of Dissimilarity.=E2=80=9D In Mathematics in the Archaeological an=
d Historical Sciences, edited by Frank Roy Hodson, David Geo=
rge Kendall, and Petre Tautu, 387=E2=80=93385. Edinburgh: Edinburgh Un=
iversity Press.=E2=80=93 Semple, Charley, and Mike Steel. 2003. Phylogenetics*. Oxford: Oxford Univ=
ersity Press.

=E2=80=93 Huson, Daniel, Regula Rupp, and Celine Scornavac= ca. 2010.

English term used throughout