Fully reducible simple Venn diagrams

TL;DR

This paper generalizes simple Venn diagrams to higher dimensions, characterizes fully reducible diagrams, and explores their properties and applications, proposing conjectures and open problems for further research.

Contribution

It introduces a framework for fully reducible simple Venn diagrams in arbitrary dimensions and provides a classification and key conditions for their reducibility.

Findings

A simple m-dimensional n-Venn diagram is fully reducible if n ≤ m+1.

Equivalent conditions for complete reducibility are established.

Conjecture: the converse of the main reducibility condition is true.

Abstract

We generalize Venn diagrams in spaces of arbitrary dimension $\geq 2$ and study simple Venn diagrams with the reducing property. Three equivalent conditions for a simple Venn diagram to reduce it completely and a classification of those diagrams is discussed. For example, we show that a simple $m$-dimensional $n$-Venn diagram is fully reducible if $n\leq m+1$ and conjecture that the converse is also true. An application of the generalized Venn diagrams and some more open problems are given.