Entropy of Closure Operators

TL;DR

This paper explores the entropy of closure operators, introduces new rank functions and axioms for matroids, and analyzes conditions for their solvability, demonstrating the density of closure entropies.

Contribution

It introduces four new rank functions, new axioms for matroids, and establishes the density of closure entropies, advancing understanding of closure operators in network coding.

Findings

Set of closure entropies is dense.

Necessary conditions for solvability of closure operators.

New axioms for matroids based on closure operators.

Abstract

The entropy of a closure operator has been recently proposed for the study of network coding and secret sharing. In this paper, we study closure operators in relation to their entropy. We first introduce four different kinds of rank functions for a given closure operator, which determine bounds on the entropy of that operator. This yields new axioms for matroids based on their closure operators. We also determine necessary conditions for a large class of closure operators to be solvable. We then define the Shannon entropy of a closure operator, and use it to prove that the set of closure entropies is dense. Finally, we justify why we focus on the solvability of closure operators only.