The intermediate logics have been classified into slices (cf. Hosoi [1]), but the detailed structure of slices has been studied only for the first two slices (cf. Hosoi and Ono [2]). In order to study the structure of slices, we give a method of a finer classification of slices & n (n 3). Here we treat only the third slice as an example, but the method can be extended to other slices in an obvious way. It is proved that each (...) subslice contains continuum of logics. A characterization of logics in each subslice is given in terms of the form of models. (shrink)

This paper deals with the varieties of monadic Heyting algebras, algebraic models of intuitionistic modal logic MIPC. We investigate semisimple, locally finite, finitely approximated and splitting varieties of monadic Heyting algebras as well as varieties with the disjunction and the existence properties. The investigation of monadic Heyting algebras clarifies the correspondence between intuitionistic modal logics over MIPC and superintuitionistic predicate logics and provides us with the solutions of several problems raised by Ono [35].

In this paper we continue the investigation of monadic Heyting algebras which we started in [2]. Here we present the representation theorem for monadic Heyting algebras and develop the duality theory for them. As a result we obtain an adequate topological semantics for intuitionistic modal logics over MIPC along with a Kripke-type semantics for them. It is also shown the importance and the effectiveness of the duality theory for further investigation of monadic Heyting algebras and logics over MIPC.