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

Cryptomorphisms for abstract rigidity matroids