Tense Logics over Lattices

Lattice theory has close connections with modal logic via algebraic semantics and lattices of modal logics. However, one less explored direction is to view lattices as relational structures based on partial orders, and study the modal logic over them. In this paper, following the earlier steps of Burgess and van Benthem in the 1980s, we use the basic tense logic and its nominal extensions with binary modalities of infimum and supremum to talk about lattices via standard Kripke semantics. As the main results, we obtain a series of complete axiomatizations of lattices, (un)bounded lattices over partial orders or strict preorders. In particular, we solve an axiomatization problem left open by Burgess (1984).