中文摘要：

在交换半环范畴中引进＂正锥＂和＂序＂等概念,实代数学中有关正锥和序的重要结果被推广到交换半环上。如下两个结果被建立：（1）一个半环是半实的,当且仅当它有一个正锥（或序）;（2）半环中一个理想为实素理想,当且仅当它是某个序的支集。

英文摘要：

The notions of positive cones and orderings are introduced in the category of commutative semirings,and some important results on cones and orderings of commutative rings are generalized to commutative semirings.Two such results are established：（1） A commutative semiring is semireal,if and only if it possesses a positive cone（or an ordering）;（2） An prime ideal of a semiring is real,if and only if it is the support of an ordering.