  In this paper, we introduce a generalization of a class of tilings which
appear in the literature: the tilings over which a height function can be
defined (for example, the famous tilings of polyominoes with dominoes). We show
that many properties of these tilings can be seen as the consequences of
properties of the generalized tilings we introduce. In particular, we show that
any tiling problem which can be modelized in our generalized framework has the
following properties: the tilability of a region can be constructively decided
in polynomial time, the number of connected components in the undirected
flip-accessibility graph can be determined, and the directed flip-accessibility
graph induces a distributive lattice structure. Finally, we give a few examples
of known tiling problems which can be viewed as particular cases of the new
notions we introduce.
