Natural duality, modality, and coalgebra

Abstract

The theory of natural dualities is a general theory of Stone–Priestley-type categorical dualities based on the machinery of universal algebra. Such dualities play a fundamental role in recent developments of coalgebraic logic. At the same time, however, natural duality theory has not subsumed important dualities in coalgebraic logic, including the Jónsson–Tarski topological duality and the Abramsky–Kupke–Kurz–Venema coalgebraic duality for the class of all modal algebras. By introducing a new notion of I S P M , in this paper, we aim to extend the theory of natural dualities so that it encompasses the Jónsson–Tarski duality and the Abramsky–Kupke–Kurz–Venema duality. The main results are topological and coalgebraic dualities for I S P M ( L ) where L is a semi-primal algebra with a bounded lattice reduct. These dualities are shown building upon the Keimel–Werner semi-primal duality theorem. Our general theory subsumes both the Jónsson–Tarski and Abramsky–Kupke–Kurz–Venema dualities. Moreover, it provides new coalgebraic dualities for algebras of many-valued modal logics and certain insights into a category-equivalence problem for categories of algebras involved. It also follows from our dualities that the corresponding categories of coalgebras have final coalgebras and cofree coalgebras. I S P M provides a natural framework for the universal algebra of modalities, and as such, for the theory of modal natural dualities.