On the structure of modal and tense operators on a boolean algebra

TL;DR

This paper investigates the algebraic and topological structure of necessity and possibility operators on boolean algebras, revealing their lattice properties and duality relations using Jonsson-Tarski duality.

Contribution

It characterizes the posets of modal and tense operators on boolean algebras as meet-semilattices and frames, with conditions for spatiality and local Stone properties.

Findings

NO(B) is a meet-semilattice that may not be distributive.

When B is complete, NO(B) forms a frame, which is spatial iff B is atomic.

Dual results apply to possibility operators and tense operators.

Abstract

We study the poset NO(B) of necessity operators on a boolean algebra B. We show that NO(B) is a meet-semilattice that need not be distributive. However, when B is complete, NO(B) is necessarily a frame, which is spatial iff B is atomic. In that case, NO(B) is a locally Stone frame. Dual results hold for the poset PO(B) of possibility operators. We also obtain similar results for the posets TNO(B) and TPO(B) of tense necessity and possibility operators on B. Our main tool is Jonsson-Tarski duality, by which such operators correspond to continuous and interior relations on the Stone space of B.