离散数学问题,求大神解答设 V = 是一个半群,若二元运算 ◦ 满足交换律,则对任意的幂等元 a, 映射fa ( x )
1个回答

首先半群满足结合律,

又因为V满足交换律,则

fa(a ◦ x)= a ◦ (a ◦ x)

=(a ◦ a) ◦ x (因为满足结合律)

=a ◦ x (因为a是幂等元) ①

fa(a)= a ◦ a =a (因为a是幂等元)

并且

fa(x)=a ◦ x

fa(a) ◦ fa(x) = a ◦(a ◦ x) =a ◦ x (由 ①式)

则 fa(a ◦ x) = fa(a) ◦ fa(x)

即f是V上的自同态。