  Hypergraph Horn functions were introduced as a subclass of Horn functions
that can be represented by a collection of circular implication rules. These
functions possess distinguished structural and computational properties. In
particular, their characterizations in terms of implicate-duality and the
closure operator provide extensions of matroid duality and the Mac
Lane-Steinitz exchange property of matroid closure, respectively.
  In the present paper, we introduce a subclass of hypergraph Horn functions
that we call matroid Horn functions. We provide multiple characterizations of
matroid Horn functions in terms of their canonical and complete CNF
representations. We also study the Boolean minimization problem for this class,
where the goal is to find a minimum size representation of a matroid Horn
function given by a CNF representation. While there are various ways to measure
the size of a CNF, we focus on the number of circuits and circuit clauses. We
determine the size of an optimal representation for binary matroids, and give
lower and upper bounds in the uniform case. For uniform matroids, we show a
strong connection between our problem and Tur\'an systems that might be of
independent combinatorial interest.
