Algebraic and coalgebraic modal logic: From Boolean algebras to semi-primal varieties

We study many-valued variants of modal logic and, more generally, coalgebraic
logic, under the assumption that the underlying algebra of truth-degrees
is a semi-primal bounded lattice-expansion. Throwing light on the category theoretical relation between the variety generated by such an algebra and
the variety of Boolean algebras, we describe multiple adjunctions between
these varieties. In particular, we show that the Boolean skeleton functor has
two adjoints, both defined by taking certain Boolean powers, and we identify
properties of these adjunctions which fully characterize semi-primality of an
algebra. Making use of these relations, we show how to lift endofunctors
encoding classical coalgebraic logics in order to obtain many-valued counterparts
of these logics. We show that one-step completeness, expressivity
and finite axiomatizability are preserved under this lifting, and we show that
for classical modal logic and similar cases, an axiomatization of the lifted
many-valued logic can be directly obtained from an axiomatization of the
original logic. Lastly, we develop the theory of natural dualities for varieties
generated by finite positive MV-chains and apply this to the algebraic study
of the negation-free fragment of bimodal finite Lukasiewicz logic.