A theory of $q$-transversals
Mark Saaltink
TL;DR
This work develops a $q$-analog of transversal theory by replacing sets with subspaces and studying $q$-transversals within a vector space. It demonstrates that partial $q$-transversals form the independent sets of a $q$-matroid, expressible as a union of rank-1 $q$-matroids and representable via matrix constructions. The paper establishes avoidance/Hall-type results, provides a cryptomorphic reformulation, and offers a representation framework including a concrete construction in a special case, while highlighting conjectures such as a $q$-Rado and questions about the minor-closed classes akin to gammoids. Overall, the work lays foundational ground for a robust $q$-transversal theory with parallels to classical transversal matroids and potential impact on representability and structural matroid theory in finite-field contexts.
Abstract
Given an indexed family ${\cal A} = (A_1, A_2, \dotsc, A_n)$ of subsets of some given set $S$, a \emph{transversal} is a set of distinct elements $x_1, x_2, \dotsc, x_n$ with each $x_i \in A_i$. Transversals have been studied since 1935 and have many attractive properties, with a deep connection to matroids. A $q$-analog is formed by replacing the notion of a set by the notion of a vector space, with a corresponding replacement of other concepts. In this paper we define a $q$-analog of the theory of transversals, and show that many of the main properties of ordinary transversals are shared by this analog.