Valuations And Real Valuations On A Semiring

In this paper, the notions of valuations and real valuations are introduced in the category of commutative semirings with units. Some basic results on valuations and real valuations of semirings are established. Moreover, the compatibility between orderings and valuations is introduced for a commutative semiring. As a necessary and sufficient condition, it is proved that a valuation is real if and only if it is compatible with some ordering.