  A variational structure for the potential AKP system is established using the
novel formalism of a Lagrangian multiforms. The structure comprises not only
the fully discrete equation on the 3D lattice, but also its semi-discrete
variants including several differential-difference equations asssociated with,
and compatible with, the partial difference equation. To this end, an overview
is given of the various (discrete and semi-discrete) variants of the KP system,
and their associated Lax representations, including a novel `generating PDE'
for the KP hierarchy. The exterior derivative of the Lagrangian 3-form for the
lattice potential KP equation is shown to exhibit a double-zero structure,
which implies the corresponding generalised Euler-Lagrange equations. Alongside
the 3-form structures, we develop a variational formulation of the
corresponding Lax systems via the square eigenfunction representation arising
from the relevant direct linearization scheme.
