Metrization of idempotent semiring of topological rings

In this paper, we have introduced natural H+ - pseudovaluation and H+ -norm on a completely regular semiring making it a topological completely regular (algebraic) semiring and established some necessary and sufficient conditions of metrizability of a topological completely regular (algebraic) semiring. We have generalized the Birkhoff-Kakutani theorem on a first countable topological completely regular (algebraic) semiring. Also we have established that a topological completely regular (algebraic) semiring is metrizable if and only if it is an M- space and its set of additive idempotents is a metrizable Gδ - set in S. Finally we have studied the completion of a b- lattice of topological rings in its uniformity structure.