同态与同构，是近世代数系统中的概念，是学习其他相关课程的基础概念。
Then the definition and properties of rough subrings, rough ideal and rough quotient rings are put forward and proved. Finally, homomorphism and isomorphism of rough rings is studied.
首先给出了粗糙环的定义及其性质，其次给出了粗糙子环、粗糙理想、粗糙商环的定义及其性质，最后研究了粗糙环的同态与同构。
参考来源 - 粗糙集理论在代数系统——群、环上的应用·2,447,543篇论文数据，部分数据来源于NoteExpress
系统数学模型的基本模式：行为模式和状态变量模式以及在状态变量模式下的同构与同态系统；
The two basic modes of mathematical model-behavior mode and state variable mode, along with the isomorphic and homomorphic relation between models and real system are described.
然后根据扩张原理，在模糊集乘积同态、同构映射的条件下，研究模糊集乘积与模糊集乘积同态像之间的关系。
Then, based on the extension principle, the relations between fuzzy sets product and the homomorphic images of the fuzzy sets product are studied.
在一元泛代数上引入了双同余关系，从而使双同态映射与双同构映射得到了沟通。
In this paper we have given the concept of double-congruence, by which the concepts of double-homomorphism and double-isomorphism in a unary algebra are linked together.
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