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Entropy of Closure Operators
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.Comment: arXiv admin note: substantial text overlap with arXiv:1209.655