  We extend the big and $p$-typical Witt vector functors from commutative rings
to commutative semirings. In the case of the big Witt vectors, this is a
repackaging of some standard facts about monomial and Schur positivity in the
combinatorics of symmetric functions. In the $p$-typical case, it uses
positivity with respect to an apparently new basis of the $p$-typical symmetric
functions. We also give explicit descriptions of the big Witt vectors of the
natural numbers and of the nonnegative reals, the second of which is a
restatement of Edrei's theorem on totally positive power series. Finally we
give some negative results on the relationship between truncated Witt vectors
and $k$-Schur positivity, and we give ten open questions.
