Logic and invariant theory IV. Invariants and syzygies in combinatorial geometry

Previous work has recast the invariant theory of projective geometry in terms of first order logic. This approach is applied to two categories connected with combinatorial projective geometry and coordinatized combinatorial pregeometries to characterize those invariant formulas (capable of expressing geometric properties) in terms of the language of brackets or determinants. The axioms for the theory of coordinatized pregeometries in this language are presented and conclusions drawn about the significance of identities or syzygies in the study of combinatorial geometry.