Higher Level Orderings In A Commutative Semiring

The notions of preordering of level n and ordering of level n are introduced in the category of commutative semirings, where n is a positive integer. Some important results on higher level 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 ordering(or a preordering) of level n, if and only if S is a semireal semiring;(2) A prime ideal of S is real, if and only if it is the support of an ordering of level n.