# 分配律

## 定義

${\displaystyle *}$${\displaystyle +}$是定义在集合${\displaystyle S}$上的兩個二元運算，我們說

• ${\displaystyle *}$对于${\displaystyle +}$满足左分配律，如果：
${\displaystyle \forall x,y,z\in S,x*(y+z)=(x*y)+(x*z)}$;
• ${\displaystyle *}$对于${\displaystyle +}$满足右分配律，如果：
${\displaystyle \forall x,y,z\in S,(y+z)*x=(y*x)+(z*x)}$;
• 如果${\displaystyle *}$对于${\displaystyle +}$同時满足左分配律和右分配律，那么我們說${\displaystyle *}$对于${\displaystyle +}$满足分配律。

## 例子

${\displaystyle \operatorname {max} (a,\operatorname {min} (b,c))=\operatorname {min} (\operatorname {max} (a,b),\operatorname {max} (a,c))}$
${\displaystyle \operatorname {min} (a,\operatorname {max} (b,c))=\operatorname {max} (\operatorname {min} (a,b),\operatorname {min} (a,c))}$
${\displaystyle \operatorname {gcd} (a,\operatorname {lcm} (b,c))=\operatorname {lcm} (\operatorname {gcd} (a,b),\operatorname {gcd} (a,c))}$
${\displaystyle \operatorname {lcm} (a,\operatorname {gcd} (b,c))=\operatorname {gcd} (\operatorname {lcm} (a,b),\operatorname {lcm} (a,c))}$
• 对于实数，加法对最大值满足分配律，对最小值也满足分配律：
${\displaystyle a+\operatorname {max} (b,c)=\operatorname {max} (a+b,a+c)}$
${\displaystyle a+\operatorname {min} (b,c)=\operatorname {min} (a+b,a+c)}$