Cryptomorphisms for abstract rigidity matroids

TL;DR

This paper advances the understanding of abstract rigidity matroids by providing multiple characterizations and proposing a conjecture about their structure, supported by partial proofs and recursive validation.

Contribution

It introduces cryptomorphic characterizations of abstract rigidity matroids and formulates a conjecture on their structural classification, with partial recursive proof.

Findings

Multiple cryptomorphic characterizations established

A conjecture on classifying rigidity matroids proposed

Recursive proof supporting the conjecture provided

Abstract

This note contributes to the structure theory of abstract rigidity matroids in general dimension. In the spirit of classical matroid theory, we prove several cryptomorphic characterizations of abstract rigidity matroids (in terms of circuits, cocircuits, bases, hyperplanes). Moreover, the study of hyperplanes in abstract rigidity matroids leads us to state (and support with significant evidence) a conjecture about characterizing the class of abstract rigidity matroids by means of certain "prescribed substructures". We then prove a recursive version of this conjecture.