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

Guram Bezhanishvili, Andre Kornell

TL;DR

The paper analyzes the structure of necessity and possibility operators on Boolean algebras, along with their tense variants, through Jónsson–Tarski duality. By translating operator posets into frames and relations on Stone spaces, it establishes when these operator sets form semilattices, frames, or Boolean algebras, and describes spatiality via atomicity. Key results show that $NO(B)$ and $TNO(B)$ are bounded meet-semilattices in general, but become frames (hence more structured) when $B$ is complete, with spatiality equivalent to atomicity; in complete-atomic cases they acquire locally Stone or completely atomic Boolean structures. Dual descriptions via continuous and interior relations provide explicit representations (e.g., through cocontinuous relations and Vietoris constructions), linking algebraic, topological, and pointfree perspectives and offering a unified framework for modal and tense operators on Boolean algebras.

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.

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

TL;DR

The paper analyzes the structure of necessity and possibility operators on Boolean algebras, along with their tense variants, through Jónsson–Tarski duality. By translating operator posets into frames and relations on Stone spaces, it establishes when these operator sets form semilattices, frames, or Boolean algebras, and describes spatiality via atomicity. Key results show that and are bounded meet-semilattices in general, but become frames (hence more structured) when is complete, with spatiality equivalent to atomicity; in complete-atomic cases they acquire locally Stone or completely atomic Boolean structures. Dual descriptions via continuous and interior relations provide explicit representations (e.g., through cocontinuous relations and Vietoris constructions), linking algebraic, topological, and pointfree perspectives and offering a unified framework for modal and tense operators on Boolean algebras.

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.
Paper Structure (4 sections, 31 theorems, 21 equations)

This paper contains 4 sections, 31 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Basic structure of $\mathcal{NO}(B)$
  3. The structure of $\mathcal{NO}(B)$ for $B$ complete
  4. The structure of $\mathcal{TNO}(B)$

Key Result

Theorem 1.1

$\sf{MA}$ is dually equivalent to $\sf{DF}$.

Theorems & Definitions (70)

  • Theorem 1.1: Jónsson-Tarski duality
  • Definition 1.2
  • Theorem 1.3: Halmos duality
  • Lemma 1.4
  • proof
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • ...and 60 more