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The system of intuitionistic modal logic ${\bf IEL}^{-}$ was proposed by S.Artemov and T. Protopopescu as the intuitionistic version of belief logic\cite{Artemov}. We construct the modal lambda calculus which is Curry-Howardisomorphic to ${\bf IEL}^{-}$ as the type-theoretical representation ofapplicative computation widely known in functional programming. We also providea categorical interpretation of this modal lambda calculus consideringcoalgebras associated with a monoidal functor on a cartesian closed category.  Finally, we study Heyting algebras and locales with corresponding operators.Such operators are used in point-free topology as well. We study compeleteKripke-Joyal-style semantics for predicate extensions of ${\bf IEL}^{-}$ andrelated logics using Dedekind-MacNeille completions and modal cover systemsintroduced by Goldblatt \cite{goldblatt2011cover}. The paper extends theconference paper published in the LFCS'20 volume \cite{rogozin2020modal}.

Categorical and Algebraic Aspects of the Intuitionistic Modal Logic $\operatorname{IEL}^{-}$ and its predicate extensions