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代数结构小结 第一部分 代数系统一般概念 一、代数系统 1．主要内容 （1）构成代数系统的成分 非空集合-----载体 集合上的二元运算和一元运算：封闭性 公理-----运算的性质 （2）二元运算的表示----表达式、运算表 （3）二元运算的性质----算律 交换、结合、幂等、消去 分配、吸收 （4）特异元素-----单位元、零元、幂等元、逆元 2．要求 （1）熟悉运算的表示法 （2）判断给定集合和运算能否构成代数系统 （3）判断给定二元运算的性质 （4）求出二元运算的特异元素 （5）了解同类型和同种代数系统的概念 3．注意：对于带参数的集合要进行讨论 二、构造代数系统 1． 主要内容---子代数、积代数和商代数 （1）子代数 定义：非空子集，关于原代数系统的运算（含 0 元运算）封闭 性质：保持所有的算律和代数常数，与原代数系统是同种的 平凡子代数 （2）积代数 定义：V1V2 性质：能保持算律(除消去律外)和代数常数，是同类型的 （3）商代数 运算定义：同余类[a]*[b]=[ab] 性质：能保持算律(除消去律以外)和代数常数，是同类型的 注意：良定义的验证 2．要求 （1）判断代数系统的子集是否构成子代数 （2）求出给定代数系统的子代数 （3）求出给定代数系统的积代数 （4）判断关系是否为同余关系 （5）求出给定代数系统的商代数 （6）掌握子代数、积代数、商代数与原代数系统的关系 三、代数系统之间的关系-----同态与同构 1．主要内容 （1）同态映射定义 f (o( x1 , x2 ,...,xk ))  o( f ( x1 ),
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