Natural duality, modality, and coalgebra

作者：Maruyama Yoshihiro^*

来源：Journal of Pure and Applied Algebra, 2012, 216(3): 565-580.

DOI：10.1016/j.jpaa.2011.07.002

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 Jonsson-Tarski topological duality and the Abramsky-Kupke-Kurz-Venema coalgebraic duality for the class of all modal algebras. By introducing a new notion of ISPM, in this paper, we aim to extend the theory of natural dualities so that it encompasses the Jonsson-Tarski duality and the Abramsky-Kupke-Kurz-Venema duality. The main results are topological and coalgebraic dualities for ISPM (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 Jonsson-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. ISPM provides a natural framework for the universal algebra of modalities, and as such, for the theory of modal natural dualities.

• 出版日期2012-3