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# 第九章 格与布尔代数 - PowerPoint PPT Presentation

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### 第九章 格与布尔代数

9.1格的定义及性质

a≼b ⇔ a|b (a整除b)。则(Z+, ≼)是格。

gcd(a,b)为a和b的最大公约数。

• 群G的全体子群S(G)对于偏序 ⊆ 构成格。

G; H∩K

• 群G的全体正规子群H(G)对于偏序 ⊆ 构成格。

H, K 的最小上界HK={ hk|h∈H, k∈K }, 最大下界H∩K

• 环R的全体理想I(R)对于偏序 ⊆ 构成格。

I, J 的最小上界I+J={ i+j|i∈I, j∈J } , 最大下界I∩J

• 线性空间V的全体子空间S(V)对于偏序⊆ 构成格。

• 设( L, ≼ )是格，由于L的任意两个元素 a和 b均有唯一的最小上界和最大下界，因此“∨”和“∧”是格中的两个二元运算，所以格可以看作具有两个二元运算“∨”和“∧”的代数系统。

1．(1) a∨a = a， (2)a∧a = a幂等律

2．(1) a∨b=b∨a， (2)a∧b=b∧a交换律

3．(1) (a∨b)∨c=a∨(b∨c), (2)(a∧b)∧c=a∧(b∧c)结合律

4．(1)a∨(a∧b)=a， (2)a∧(a∨b)=a 吸收律

(a∨b)∨c=a∨(b∨c)

x ∘y= x ∨y，x ∗y= x ∧y

a∧b = glb(a,b) = a*b

a∨b =lub(a, b) = a∘b