Duality and tangles of set separations

TL;DR

This paper investigates a duality in connectivity systems involving tangles, establishing a correspondence between two sets and their associated subsets, and explores how tangles relate across these dual systems.

Contribution

It introduces a formal duality framework for tangles in connectivity systems and analyzes the relationship between tangles in dual structures.

Findings

Established a duality between sets in connectivity systems.

Linked tangles in dual systems through subset correspondences.

Provided insights into the structure of tangles via duality.

Abstract

Applications of tangles of connectivity systems suggest a duality between these, in which for two sets $X$ and $Y\!$ the elements $x$ of $X$ map to subsets $Y_x$ of $Y\!$, and the elements $y$ of $Y\!$ map to subsets $X_y$ of $X$, so that $x\in X_y$ if and only if $y\in Y_x$ for all $x\in X$ and $y\in Y\!$. We explore this duality, and relate the tangles arising from the dual systems to each other.