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In chapter 1, we simply introduce the background and the present state of the theory of semirings; for the sake of convenience, we introduce some fundamental knowledge about idempotent semirings.

第一章，我们对半环理论研究的背景、现状和幂等元半环的基础知识作了简要的介绍。

Ce permits you to mark individual operations as idempotent.

CE允许你将单个的的操作标记为幂等操作。

A sub variety of idempotent semiring variety which satisfies the additional identity and is given in this paper.

给出了满足附加等式x+xy+x≈x+y，x+xy≈xy的幂等元半环簇 I的一个子簇，讨论了该簇中成员的一些性质，并得到了这些性质的几个等价刻画和其成员的Mal'cev积分解。

We show several necessary and sufficient conditions for the invertibility of commutator of idempotent and involutory matrices.

研究了复数域上两个n阶幂等矩阵P与Q的换位子PQ-QP的可逆性问题。

Idempotent operations can be safely executed more than once.

改译：幂等操作可以多於一次安全执行。

The class of idempotent semirings is a very important class of semirings. Many experts and scholars thorough, painstakingly and systematically investigate it.

幂等元半环是一类非常重要的半环，许多专家学者对其进行了深入细致的系统研究。

A class of cyclic codes and their duals are also discussed, and the idempotent generators of this class of cyclic codes and their duals are described.

讨论了R上一类循环码及其对偶码，并给出了这类循环码及其对偶码的幂等生成元。

In this paper we mainly study some important classes of idempotent semirings. First, we give certain characterizations of the members of them from all kinds of angles; second, we obtain the subdirect product decomposition of some semirings which are members of them.

本文主要研究了几个重要的幂等元半环簇，从多个角度给出了其中成员的刻划，也对他们进行了次直积分解。

R/PE is a pre-division ring. P is any prime ideal in R, E is a set of all idempotent elements of P.

3R/PE为准体，其中P为R的任意素理想，E为P的所有幂等元素组成的集合。

One construction of Cartesian authentication codes from the normalform of Idempotent matrices over the field Fq is presented.

本文利用有限域上幂等阵的相似标准形，构造了一个Cartesian认证码，并计算出该码的所有参数。

Breath, muscle contraction of the buttocks; arch body, as far as possible to hold his head, right leg straight towards the ceiling (peg-leg knee in order to avoid muscle tension).

呼气，收缩臀部肌肉；拱起身体，尽量抬起头来，右腿伸直朝向天花板（膝微屈，以避免肌肉紧张）。

The cost of moving grain food products was unchanged from May, but year over year are up 8%.

粮食产品的运输费用与5月份相比没有变化，但却比去年同期高8％。