# 單位元

提示：呢篇文講嘅唔係單位圓

## 定義

${\displaystyle (S,*)}$係一個有二元運算${\displaystyle *}$嘅集合${\displaystyle S}$（稱之為原羣），若果對任何S入面嘅b都有${\displaystyle e_{1}*b=b}$，咁${\displaystyle e_{1}}$就叫做一個左單位元；若果對任何S入面嘅a都有${\displaystyle a*e_{2}=a}$，咁${\displaystyle e_{2}}$就叫做一個右單位元。若果${\displaystyle e}$同時係左單位元同右單位元，則叫雙邊單位元，亦都可以簡稱單位元

## 例子

${\displaystyle e*e=f*e=e}$${\displaystyle f*f=e*f=f}$

${\displaystyle e}$${\displaystyle f}$都係左單位元，但不存在右單位元或者雙邊單位元

## 參考資料

### 書目

• Beauregard, Raymond A.; Fraleigh, John B. (1973), A First Course In Linear Algebra: with Optional Introduction to Groups, Rings, and Fields, Boston: Houghton Mifflin Company, ISBN 0-395-14017-X
• Fraleigh, John B. (1976), A First Course In Abstract Algebra (第2版), Reading: Addison-Wesley, ISBN 0-201-01984-1
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016
• McCoy, Neal H. (1973), Introduction To Modern Algebra, Revised Edition, Boston: Allyn and Bacon, LCCN 68015225

## 讀埋

• M. Kilp, U. Knauer, A.V. Mikhalev, Monoids, Acts and Categories with Applications to Wreath Products and Graphs, De Gruyter Expositions in Mathematics vol. 29, Walter de Gruyter, 2000, ISBN 3-11-015248-7, p. 14–15