Tractable Combinations of Temporal CSPs

The constraint satisfaction problem (CSP) of a first-order theory T is the computational deciding whether given conjunction atomic formulas satisfiable in some model T. We study complexity CSP$(T_1 \cup T_2)$ where $T_1$ and $T_2$ are theories with disjoint finite relational signatures. prove that if temporal structures, i.e., structures all relations have definition $(Q;<)$, then P or NP-complete. To this end we purely algebraic statement about structure lattice locally closed clones over domain $Q$ contain Aut$(Q;<)$.