Definability and decidability of binary predicates for time granularity

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