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On Strictly Positive Modal Logics with S4.3 Frames

We investigate the lattice of strictly positive (SP) modal logics that contain the SP-fragment of the propositional modal logic S4.3 of linear quasi-orders. We are interested in Kripke (in)completeness of these logics, their computational complexity, as well as the definability of Kripke frames by means of SP-implications. We compare the lattice of these SP-logics with the lattice of normal modal logics above S4.3. We also consider global consequence relations for SP-logics, focusing on definability and Kripke completeness.