A theory of $q$-transversals

Abstract

Given an indexed family $\mathcal{A}=(A_1,A_2,\dots ,A_n)$ of subsets of some given set $S$, a \emph{transversal} is a set of distinct elements $x_1,x_2,\dots ,x_n$ 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.