On Invariants and Cyclic Flat Structures in the Theory of q-Matroids and L-Polymatroids

This thesis investigates generalisations of core matroid theory concepts to the settings of q-matroids and L- polymatroids. In particular, we focus on the Tutte polynomial of a q-matroid, the free product of q-matroids, and the cyclic flats of L-polymatroids. A central theme throughout the thesis is the concept of cyclic flats, which underpins our approach in each instance of generalisation. We generalise the Tutte polynomial of a matroid to the q-matroid setting by considering q-matroid minors that contain a singular cyclic flat. With this approach, we establish a notion of the Tutte polynomial for all q-matroids whose support lattice admits a proper interval decomposition. Moreover, we establish an invertible convolution transformation between the Tutte polynomial and the rank generating polynomial of a q-matroid. In addition, we generalise the binary operation of the free product from matroid theory to the q-matroid setting. We first establish this generalisation through the lens of the rank function and independent spaces, each of which deviates significantly from the classical case. We then provide a characterisation of the free product via the lattice of cyclic flats, which permits a more elegant description of this operation. In particular, this characterisation via the lattice of cyclic flats allows us to establish fundamental properties of the free product of q-matroids, including unique factorisation and maximality in the weak order. Moreover, we establish initial results on representability. Finally, we extend the notion of cyclic flats beyond the q-matroid and polymatroid settings to the setting of L-polymatroids. On complemented modular lattices, we provide an axiomatisation of the cyclic flats of an L-polymatroid. We introduce the notion of a join-decomposition function, which plays a fundamental role in determining an L-polymatroid from a weighted lattice Z embedded in a complemented modular lattice L, together with an atomic weighting of L. The results presented here contribute to the emerging landscape of q-analogue combinatorics, providing new tools and perspectives for understanding rank-metric codes, cyclic flat structures, and algebraic invariants beyond the Boolean setting.