  In this paper we develop a number of results and notions concerning
Positivstellens\"atze for semirings (preprimes) of commutative unital real
algebras. First we reduce the Archimedean Positivstellensatz for semirings to
the corresponding result for quadratic modules. Various applications of the
Archimedean Positivstellensatz for semirings are investigated. A general
Positivstellensatz with denominators is proved for filtered algebras with
semirings. As an application we derive a denominator-free Positivstellensatz
for the cylindrical extension of an algebra with Archimedean semiring. A large
number of illustrating examples are given.
