  It is well-known that small categories have equivalent descriptions as
partial monoids. We provide a formulation of partial monoid and partial monoid
homomorphism involving $s$ and $t$ instead of identities and then following a
recent investigation into involutive double categories, we prove that a double
category is equivalent to a set equipped with two partial monoid structures in
which all structure maps $(s,t,\circ)$ are partial monoid homomorphisms. We
discuss the light this purely algebraic perspective sheds on the symmetries of
these structures and its applications. Iteration of the procedure leads also to
a pure algebraic formulation of the notion of $n$-fold category.
