  Cross-connection is a construction of regular semigroups using certain
categories called normal categories which are abstractions of the partially
ordered sets of principal left (right) ideals of a semigroup. We describe the
cross-connections in the semigroup $Sing(X)$ of all non-invertible
transformations on a set $X$. The categories involved are characterized as the
powerset category $\mathscr{P}(X)$ and the category of partitions $\Pi(X)$. We
describe these categories and show how a permutation on $X$ gives rise to a
cross-connection. Further we prove that every cross-connection between them is
induced by a permutation and construct the regular semigroups that arise from
the cross-connections. We show that each of the cross-connection semigroups
arising this way is isomorphic to $Sing(X)$. We also describe the right
reductive subsemigroups of $Sing(X)$ with the category of principal left ideals
isomorphic to $\mathscr{P}(X)$. This study sheds light into the more general
theory of cross-connections and also provides an alternate way of studying the
structure of $Sing(X)$.
