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设S是一个右单纯、右可消的半群。Let S be a right simple right cancellative semigroup.

给出该类半群的一个构造定理。A construction theorem for such semigroup is obtained.

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为右群强半格的诣零理想扩张。S is a nil-extension of strong semilattice of right semigroup.

证明了一个半群上所有模糊同余关系作成一个格。We show that all fuzzy congruence relations on a semigroup is a lattice.

交换半群的极小理想如果存在的话是个群。The minimal ideal of a commutative semigroup , when it exists, is a group.

半群被称为周期性的，如果所有它的元素有着有限次序。A semigroup is said to be periodic if all of its elements are of finite order.

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.

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文中将进一步讨论该半群的性质，证明该半群是不可约的。This paper is a further effort to investigate the irreducibility of this semigroup.

在第四节，我们针对迁移性定义对群无意义的事实定义了半迁移线性半群。The concept of semi-transitive linear semigroup is introduced in the fourth section.

本文主要研究超富足半群的平移壳问题。This paper mainly focuses on the study of the translational hull of superabundant semigroup.

没有极小理想的半群的例子是在加法下的正整数集合。An example of semigroup with no minimal ideal is the set of positive integers under addition.

称半群S为2-半带，若其中每个元素都可以写为S中两个幂等元的积。A semigroup S is called 2-semiband, if every element of S is a product of two idempotents of S.

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类似于群分次环和群分次模的定义，定义了部分半群分次环和部分半群分次模。Similar to group graded rings and modules, partial semigroup graded rings and modules are defined.

最后，利用算子半群理论，证明了形状记忆合金模型局部解的存在唯一性。Finally, by applying the semigroup theory, the existence and uniqueness of local solution are proved.

设S是有向序半群，本文给出了S上的一类正则同余，称为强序同余的定义及性质。In this paper, the notation of a strongly ordered congruence on an ordered semigroup S is introduced.

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.

特别地，还定义了半群的左右反同态并讨论了他们的性质。Especially, It also defined left and right anti-homorphism on semigroup and discussed their properties.

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最后我们利用序模糊点和模糊双理想对正则舵序半群给出了刻画。Finally, we characterize regular duo ordered semigroup S by ordered fuzzy points and fuzzy bi-ideals of S.

左半正规纯正半群是幂等元集形成左半正规带的纯正半群。A left seminormal orthodox semigroup is an orthodox semigroup whose idempotents form a left seminormal band.

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.