TL;DR
This paper extends the theory of infinite matroids by generalizing modular pairs, developing new techniques, and exploring properties like extensions, quotients, and connectivity, with formalized proofs available online.
Contribution
It introduces a comprehensive framework for infinite matroids, including single-element extensions, relationships between quotients and projections, and new insights into connectivity.
Findings
Complete theory of single-element extensions
Infinite matroids with all flats modular are finitary
New perspectives on the infinite matroid connectivity parameter
Abstract
We generalize the well-studied notion of a modular pair of a finite matroid to arbitrary families of sets in infinite matroids, and use it to develop the theory of infinite matroids in several as-yet-unexplored areas. Our results include a complete theory of single-element extensions, a description of the relationship between quotients and projections, a proof that matroids for which every flat is modular must be finitary, and two new perspectives on the infinite matroid connectivity parameter \lambda. In most cases, existing theory for finite matroids either fails completely or does not extend in obvious ways, and as a result we develop multiple new techniques for reasoning about infinite matroids, including establishing well-behaved infinite analogues of nullity, local connectivity and skewness. We also point to an online repository containing formalized proofs of all our results…
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