The Logic of Partitions and Partition Logics: Ore's Correspondence, Contextual Pasting, and Direct-Sum Decompositions

2026-08-09

AI Summary

This work aims to unify partition logics across different levels by clarifying the relationship between Boolean event algebras generated by partitions and their global structure, drawing an analogy with direct-sum decompositions of vector spaces. By embedding partitions of finite sets into a Boolean algebraic framework via the Ore correspondence and characterizing their algebraic properties through context-dependent pasting operations, the study introduces orthogonal direct-sum decompositions as a natural generalization of partitions in quantum contexts. It reveals fundamental distinctions among shared events, atomic intertwining, and inherited order structures within partition logic and establishes precise correspondences between partition operations and Boolean meet and join. This framework provides a unified logical foundation for automata models, generalized urn models, and quantum measurements, including foundational results such as Gleason’s and the Kochen–Specker theorems.

Abstract

The term ``partition logic'' denotes two constructions at different levels. In automaton and generalized-urn models, selected partitions generate Boolean event algebras whose contextwise union forms a concrete pasted event structure; in Ellerman's framework, whole partitions are classifications governed by refinement and partition operations. For a finite set $U$, Ore's correspondence maps each generator $π$ to its Boolean algebra $\BA(π)$, but it neither identifies the pasted carrier with $\Part(U)$ nor makes pasting a partition operation. It yields $\BA(π\wedgeσ)=\BA(π)\cap\BA(σ)$ and $\BA(π\veeσ)=\langle\BA(π)\cup\BA(σ)\rangle_{\rm BA}$, where $\langle\cdot\rangle_{\rm BA}$ denotes Boolean-algebra generation. Thus meet captures the common event algebra, whereas join gives the ambient Boolean closure. Chinese-lantern, Firefly, and triangular examples distinguish shared events, atomic intertwining, and inherited concrete order. Ellerman's direct-sum decompositions (DSDs) provide a vector-space analogue: component projections of an orthogonal DSD resolve the identity and encode exclusive outcomes, but its components are not equivalence classes of vectors. Gleason and Kochen--Specker applications require globally context-consistent valuations on those projections.

Problem

• partition logic
• Ore's correspondence
• contextual pasting
• direct-sum decompositions
• Boolean event algebras

Innovation

• partition logic
• Ore's correspondence
• direct-sum decomposition
• Boolean event algebra
• contextual pasting