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首先利用弱理想，引入了弱全幂等环的概念。The conception of weak completely idempotent rings is introduced.

最后还证明了幂等算子和它的伴随是相似的。In the end we prove that idempotent operator is similar to its adjoint.

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如果它是有限和非空的，则它必须包含至少一个幂等元。If it is finite and nonempty , then it must contain at least one idempotent.

得出了所有非空周期半群都有至少一个幂等元。It follows that every nonempty periodic semigroup has at least one idempotent.

从设计的角度看，这也意味着消息处理的次数会幂等增加。From a design standpoint this means that message processing has to be idempotent.

由于并发相关操作的失败而导致失败的等幂业务服务可以重试。An idempotent business service that fails with a concurrency related failure can be retried

每个子群精确的包含一个幂等元，也就是这个子群的单位元。Each subgroup contains exactly one idempotent , namely the identity element of the subgroup.

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正则半群上的同余是由其幂等元同余类所完全决定的。The congruences on a regular semigroup is completely determined by its idempotent congruence classes.

坡是个加法幂等，乘法小于或等于其因子的半环。Inclines are additively idempotent semi rings in which products are less than or equal to either factor.

例如，所有非空有限半群是周期性的，并有一个极小理想和至少一个幂等元。For example, every nonempty finite semigroup is periodic, and has a minimal ideal and at least one idempotent.

每一个主左理想均由一个幂等元生成的环叫正则环。A associative ring R is called a regular ring if every left principle ideal of it is generated by an idempotent.

第二个问题是以幂等方式处理序列消息，这样这些消息被执行一次，且一次。A second problem was handling sequential messages in an idempotent fashion, such that they are executed once and only once.

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刻画了弱逆半群S上的最大幂等元分离同余和最小群同余。In this paper, the greatest idempotent separating congruence and the minimum group congruence on a weakly inverse semigroup S are characterized.

由此推出了P-正则半群上的每个P-同余完全是由其包含幂等元的部分核正规系所决定的。So We have prove that each P-congruence on P-regular semigroups is uniquely determined by its partial kernel normal systems containing idempotent elements.

特别地，我们给出了严格单锥形幂等元剩余格的一个分类。Finally, based on the structure theorem, we obtain some characterizations of subdirectly irreducible, simple and strictly simple conical idempotent residuated lattices.

第一章、第二章分别利用幂等矩阵及反对称矩阵的标准型构造了卡氏认证码。Chapter one and chapter two use normal form of idempotent matrices and antisymmetric matrices over finite fields to construct cartesian authentication codes respectively.

本文进一步推广上述分解方法到一类满足正则性条件且含最大寨等元的自然序富足半群。In this paper, we further extend that method to a class of naturally ordered abundant semigroups which satisfies the regularity condition and contains a greatest idempotent.

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本文证明在类空间存在一生成基，与投影算符或幂等元相反，它是各次幂不等的。In this paper we showed that there is a generating basis in class space. It differs from the projection operators and idempotent for every power which is not equal to itself.

本文从研究矩形的双序集性质入手，推导出矩形带半群与它的幂等元双序集，即矩形的双序集之间的一些重要关系。This paper proves some important properties of rectangular biordered set and shows relations between rectangular band and its idempotent set which is a rectangular biordered set.

给出了布尔群代数半群中的幂等元、极大子群和正则元的结构以及幂等元和正则元的个数。The structure of the idempotent elements, regular elements and maximal subgroups and the number of the idempotent elements and regular elements in Boolean group algebra are given.