  Phylogenetic networks are a generalization of evolutionary or phylogenetic
trees that are commonly used to represent the evolution of species which cross
with one another. A special type of phylogenetic network is an {\em
$X$-cactus}, which is essentially a cactus graph in which all vertices with
degree less than three are labelled by at least one element from a set $X$ of
species. In this paper, we present a way to {\em encode} $X$-cactuses in terms
of certain collections of partitions of $X$ that naturally arise from
$X$-cactuses. Using this encoding, we also introduce a partial order on the set
of $X$-cactuses (up to isomorphism), and derive some structural properties of
the resulting partially ordered set. This includes an analysis of some
properties of its least upper and greatest lower bounds. Our results not only
extend some fundamental properties of phylogenetic trees to $X$-cactuses, but
also provides a new approach to solving topical problems in phylogenetic
network theory such as deriving consensus networks.
