Higher level orderings in commutative semiring

The notions of preorderings of level n and orderings of level n are introduced in the category of commutative semirings,where n is a positive integer.Some important results on higher orderings of commutative rings are generalized to commutative semirings.For a commutative semiring S and an arbitrary positive integer n,two such results are established:(1) S possesses an orderings of level n,if and only if S is a semireal semiring;(2) An prime ideal of S is real,if and only if it is the support of an ordering of level n.