半环的高层序

Higher level orderings in commutative semiring

在交换半环范畴中引进"n层亚序"和"n层序"等概念,其中n为任意正整数。实代数学中有关高层序的重要结果被推广到交换半环上。对于一个半环S以及任意给定的正整数n,两个这样的结果被建立:(1)半环S有一个n层序,当且仅当S是半实的;(2)半环S中一个理想为实素理想,当且仅当它是某个n层序的支集。

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 com- mutative 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.