We extend the notion of matroid representations by matrices over fields and
consider new representations of matroids by matrices over finite semirings,
more precisely over the boolean and the superboolean semirings. This idea of
representations is generalized naturally to include also hereditary
collections. We show that a matroid that can be directly decomposed as
matroids, each of which is representable over a field, has a boolean
representation, and more generally that any arbitrary hereditary collection is
superboolean-representable.Comment: 27 page
