Idempotent compatible maps and discrete integrable systems on the triangular lattice

We present three equivalence classes of rational non-invertible multidimensional compatible maps. These maps turns out to be idempotent and by construction they admit birational partial inverses (companion maps) which are Yang-Baxter maps. The maps in question can be reinterpreted as systems of difference equations defined on the edges of the $\mathbb{Z}^2$ graph. Finally, we associate these compatible systems of difference equations with integrable difference equations defined on the triangular lattice $Q(A2)$.