# A Graph-theoretic Method to Define any Boolean Operation on Partitions

## TL;DR

This paper introduces a novel graph-theoretic approach to define any Boolean operation on set partitions, extending classical lattice operations with a natural and systematic method.

## Contribution

It presents a new graph-theoretic and closure-theoretic framework for defining all n-ary Boolean operations on partitions, filling a historical gap.

## Key findings

- A simple graph-based method for Boolean operations on partitions
- An equivalent closure-theoretic approach established
- Addresses historical delay in defining all Boolean operations

## Abstract

The lattice operations of join and meet were defined for set partitions in the nineteenth century, but no new logical operations on partitions were defined and studied during the twentieth century. Yet there is a simple and natural graph-theoretic method presented here to define any n-ary Boolean operation on partitions. An equivalent closure-theoretic method is also defined. In closing, the question is addressed of why it took so long for all Boolean operations to be defined for partitions.