2017/07/11 by Guus P. Bollen, Bollen, Guus, Henry Crapo +3 · 1 citation
Computer Science · Engineering · Mathematics · #05B35 #Advanced Algebra and Logic #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1707.03459
openalex publication_date 2017/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
q-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials τ(x,y) of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution x → (x-1), y → (y-1) in their rank generating functions ρ(x,y). The q-analogue of the passage from a Tutte polynomial to its corresponding RGF is straight-forward, but the analogue of the reverse process x → (x-1), y → (y-1) is more delicate. For matroids M(S) on a set S, and relative to any linear order on the points, the concept of internal/external activity of a point relative to a basis gives rise to a partition of the underlying Boolean algebra B(S) into a set of "prime-free" (or "structureless") minors, such minors being direct sums of loops and isthmi (coloops), with one such prime-free minor for each basis. What usually goes unnoticed is that each prime-free minor has a unique clopen flat. The latter property carries over to q-matroids, but each prime-free minor will contain many bases. So internal and external activity in q-matroids must be defined not for points relative to bases, but rather for coverings in the underlying complemented modular lattice. Following lattice paths from arbitrary subspaces A along active coverings (downward for internally active, upward for externally active) will lead to the unique clopen subspace in the prime-free minor containing the subspace A. There are a number of interesting questions concerning q-matroids that remain unsolved.