Observations relating to the equivalences induced on model sets by bidirectional transformations

A bidirectional transformation on a pair of sets of models induces two principal equivalence relations on each set of models. Since a model can be uniquely identified by specifying its equivalence class in each of these relations, they function as a coordinate system for the model sets, with respect to the transformation. We prove some results relating to this observation. Using them we give the implication relationships between various properties of bidirectional transformations. In particular, we characterise the bidirectional transformations that can be decomposed into a pair of lenses working “tail to tail”.