  In this paper, first we introduce dense unital magmas and magma-valued metric
spaces. The density property of the ordered unital magmas, monoids, and
hemirings helps us to generalize a couple of classical results related to the
convergence and Cauchy property of sequences and series. For example, we prove
that the Cauchy Condensation Test holds for Cauchy complete fields. We also
introduce hemiring-valued pseudonormed rings which are a generalization of
pseudonormed rings and valuation domains. Then, we prove that
finite-dimensional algebras over hemiring-valued pseudonormed fields can be
pseudonormed which is a generalization of a result by Albert.
