Overview
This research investigates the algebraic properties of profinite interior algebras, specifically focusing on their categorical characteristics. The primary objective is to identify which normal modal logics $L$, particularly those extending $S4$, lead to profinite $L$-algebras possessing properties such as finite coproducts, coregularity, and coexactness.
Research Context
Prior work established that the category of profinite $L$-algebras, where $L$ is a normal modal logic characterized by the finite model property, is monadic over $\textbf{Set}$. This earlier research also involved the development of sequent calculi designed for extensions of the language of $L$ to include infinitary conjunctions and disjunctions. The completeness of these calculi with respect to profinite $L$-algebras was demonstrated. Furthermore, these previous studies explored the relationship between syntactic properties of the developed sequent calculi and the regularity or exactness properties observed in the category opposite to that of profinite $L$-algebras.
Approach
The current study adopts an algebraic perspective. It aims to directly characterize the normal modal logics $L$ that extend $S4$. The characterization is based on whether the corresponding profinite $L$-algebras exhibit specific categorical properties: finite coproducts, coregularity, and coexactness. This approach focuses on the algebraic structure itself rather than the syntactic properties of associated sequent calculi.
Findings
- The research characterizes specific normal modal logics $L$ (extending $S4$) for which their profinite algebras possess finite coproducts.
- It further characterizes those $L$ (extending $S4$) whose profinite algebras exhibit coregularity.
- The study also identifies the conditions under which the profinite algebras for $L$ (extending $S4$) display coexactness.