Weak Lax pairs for lattice equations

We consider various 2D lattice equations and their integrability, from the point of view of 3D consistency, Lax pairs and Bäcklund transformations. We show that these concepts, which are associated with integrability, are not strictly equivalent. In the course of our analysis, we introduce a number of black and white lattice models, as well as variants of the functional Yang–Baxter equation.