Definability and decidability of binary predicates for time granularity

In this paper, we study the definability and decidability of binary predicates for time granularity with respect to monadic theories over finitely and infinitely layered structures. We focus our attention on the equi-level (resp. equi-column) predicate constraining two time points to belong to the same layer (resp. column) and on the horizontal (resp. vertical) successor predicate relating a time point to its successor within a given layer (resp. column). We give a number of positive and negative results by reduction to/from a wide spectrum of decidable/undecidable problems.