On Modal Products with the Logic of 'Elsewhere'

The finitely axiomatisable and decidable modal logic Diff of ‘elsewhere’ (or ‘difference operator’) is known to be quite similar to S5. Their validity problems have the same coNP complexity, and their Kripke frames have similar structures: equivalence relations for S5, and ‘almost’ equivalence relations, with the possibility of some irreflexive points, for Diff. However, their behaviour may differ dramatically as components of two-dimensional logics. Here we consider the decision problems of modal product logics of the form L × Diff. We present some cases where the transition from L × S5 to L × Diff not only increases the complexity of the validity problem, but in fact introduces undecidability, sometimes even non-recursive enumerability