We investigate valuated matroids with an additional algebraic structure on
their residue matroids. We encode the structure in terms of representability
over stringent hyperfields.
A hyperfield H is {\em stringent} if aβb is a singleton unless
a=βb, for all a,bβH. By a construction of Marc Krasner, each valued
field gives rise to a stringent hyperfield.
We show that if H is a stringent skew hyperfield, then the vectors of any
weak matroid over H are orthogonal to its covectors, and we deduce that weak
matroids over H are strong matroids over H. Also, we present vector axioms
for matroids over stringent skew hyperfields which generalize the vector axioms
for oriented matroids and valuated matroids.Comment: 19 page