Characterizing intermediate tense logics in terms of Galois connections

We propose a uniform way of defining for every logic L intermediate between intuitionistic and classical logics, the corresponding intermediate tense logic LKt. This is done by building the fusion of two copies of intermediate logic with a Galois connection LGC, and then interlinking their operators by two Fischer Servi axioms. The resulting system is called L2GC + FS. In the cases of intuitionistic logic Int and classical logic Cl, it is noted that Int2GC + FS is syntactically equivalent to intuitionistic tense logic IKt by W. B. Ewald and Cl2GC + FS equals classical tense logic Kt. This justifies calling L2GC + FS the L-tense logic LKt for any intermediate logic L. We define H2GC + FS-algebras as expansions of HK1-algebras, introduced by E. Orłowska and I. Rewitzky. For each intermediate logic L, we show algebraic completeness of L2GC + FS and its conservativeness over L. We prove relational completeness of Int2GC + FS with respect to the models defined on IK-frames introduced by G. Fischer Servi. We also prove a representation theorem stating that every H2GC + FS-algebra can be embedded into the complex algebra of its canonical IK-frame.