The smallest class of binary matroids closed under direct sums and complements

TL;DR

This paper introduces binary comatroids, a class generated from the empty matroid using direct sums and complements, and characterizes binary non-comatroids with all proper flats being comatroids.

Contribution

It defines binary comatroids analogous to cographs and characterizes binary non-comatroids with all proper flats as comatroids, extending to ternary matroids.

Findings

Proper flats of binary comatroids are binary comatroids.

Characterization of binary non-comatroids with all proper flats as comatroids.

Extension of results to ternary matroids.

Abstract

The class of cographs or complement-reducible graphs is the class of graphs that can be generated from $K_1$ using the operations of disjoint union and complementation. By analogy, this paper introduces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.