本文所有內容來自上海科學技術文獻出版社《離散數學》第三篇。
定義 5 - 1.1 A A A A n A^n A n B B B A A A n n n 運算 。如果 B ⊆ A B\subseteq A B ⊆ A n n n 封閉的 。
定義 5 - 1.2 A A A f 1 , f 2 , ⋯  , f k f_1,f_2,\cdots,f_k f 1 , f 2 , ⋯ , f k 代數系統 ，記作 ⟨ A , f 1 , f 2 , ⋯  , f k ⟩ \langle A,f_1,f_2,\cdots,f_k\rangle ⟨ A , f 1 , f 2 , ⋯ , f k ⟩
定義 5 - 2.1 ∗ * ∗ A A A x , y ∈ A x,y\in A x , y ∈ A x ∗ y ∈ A x*y\in A x ∗ y ∈ A ∗ * ∗ A A A 封閉的 。
定義 5 - 2.2 ∗ * ∗ A A A x , y ∈ A x,y\in A x , y ∈ A x ∗ y = y ∗ x x*y=y*x x ∗ y = y ∗ x ∗ * ∗ A A A 可交換的 。
定義 5 - 2.3 ∗ * ∗ A A A x , y , z ∈ A x,y,z\in A x , y , z ∈ A ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) (x*y)*z=x*(y*z) ( x ∗ y ) ∗ z = x ∗ ( y ∗ z ) ∗ * ∗ A A A 可結合的 。
定義 5 - 2.4 ∗ , △ *,\triangle ∗ , △ A A A x , y , z ∈ A x,y,z\in A x , y , z ∈ A x ∗ ( y △ z ) = ( x ∗ y ) △ ( x ∗ z ) ( y △ z ) ∗ x = ( y ∗ x ) △ ( z ∗ x ) x*(y\triangle z)=(x*y)\triangle(x*z)\\(y\triangle z)*x=(y* x)\triangle(z*x) x ∗ ( y △ z ) = ( x ∗ y ) △ ( x ∗ z ) ( y △ z ) ∗ x = ( y ∗ x ) △ ( z ∗ x ) ∗ * ∗ △ \triangle △ 可分配的 。
定義 5 - 2.5 ∗ , △ *,\triangle ∗ , △ A A A x , y ∈ A x,y\in A x , y ∈ A x ∗ ( x △ y ) = x x △ ( x ∗ y ) = x x*(x\triangle y)=x\\x\triangle (x*y)=x x ∗ ( x △ y ) = x x △ ( x ∗ y ) = x ∗ * ∗ △ \triangle △ 吸收律 。
定義 5 - 2.6 ∗ * ∗ A A A x ∈ A x\in A x ∈ A x ∗ x = x x*x=x x ∗ x = x ∗ * ∗ 等冪的 。
定義 5 - 2.7 ∗ * ∗ A A A e l ∈ A e_l\in A e l ∈ A x ∈ A x\in A x ∈ A e l ∗ x = x e_l*x=x e l ∗ x = x e l e_l e l A A A ∗ * ∗ 左幺元 ；如果有一個元素 e r ∈ A e_r\in A e r ∈ A x ∈ A x\in A x ∈ A x ∗ e r = x x*e_r=x x ∗ e r = x e r e_r e r A A A ∗ * ∗ 右幺元 ；如果 A A A e e e e e e A A A ∗ * ∗ 幺元 。
定理 5 - 2.1 ∗ * ∗ A A A A A A ∗ * ∗ e l e_l e l e r e_r e r e l = e r = e e_l=e_r=e e l = e r = e A A A
定義 5 - 2.8 ∗ * ∗ A A A θ l ∈ A \theta_l\in A θ l ∈ A x ∈ A x\in A x ∈ A θ l ∗ x = θ l \theta_l*x=\theta_l θ l ∗ x = θ l θ l \theta_l θ l A A A ∗ * ∗ 左零元 ；如果有一個元素 θ r ∈ A \theta_r\in A θ r ∈ A x ∈ A x\in A x ∈ A x ∗ θ r = θ r x*\theta_r=\theta_r x ∗ θ r = θ r θ r \theta_r θ r A A A ∗ * ∗ 右零元 ；如果 A A A θ \theta θ θ \theta θ A A A ∗ * ∗ 零元 。
定理 5 - 2.2 ∗ * ∗ A A A A A A ∗ * ∗ θ l \theta_l θ l θ r \theta_r θ r θ l = θ r = θ \theta_l=\theta_r=\theta θ l = θ r = θ A A A
定理 5 - 2.3 ⟨ A , ∗ ⟩ \langle A,* \rangle ⟨ A , ∗ ⟩ A A A 1 1 1 e e e θ \theta θ θ ≠ e \theta \ne e θ ̸ = e
證明： θ = e \theta=e θ = e x ∈ A x\in A x ∈ A x = e ∗ x = θ ∗ x = θ = e x=e*x=\theta*x=\theta=e x = e ∗ x = θ ∗ x = θ = e A A A
定義 5 - 2.9 ⟨ A , ∗ ⟩ \langle A,* \rangle ⟨ A , ∗ ⟩ ∗ * ∗ A A A e e e A A A ∗ * ∗ A A A a a a A A A b b b b ∗ a = e b*a=e b ∗ a = e b b b a a a 左逆元 ；如果 a ∗ b = e a*b=e a ∗ b = e b b b a a a 右逆元 ；如果一個元素 b b b a a a a a a b b b a a a 逆元 。記一個元素 x x x x − 1 x^{-1} x − 1
定理 5 - 2.4 ⟨ A , ∗ ⟩ \langle A,* \rangle ⟨ A , ∗ ⟩ ∗ * ∗ A A A A A A e e e ∗ * ∗
證明： a , b , c ∈ A a,b,c\in A a , b , c ∈ A b b b a a a c c c b b b e = c ∗ b = c ∗ ( ( b ∗ a ) ∗ b ) = ( ( c ∗ b ) ∗ a ) ∗ b = a ∗ b e=c*b=c*((b*a)*b)=((c*b)*a)*b=a*b e = c ∗ b = c ∗ ( ( b ∗ a ) ∗ b ) = ( ( c ∗ b ) ∗ a ) ∗ b = a ∗ b b b b a a a a a a b , c b,c b , c b = b ∗ e = b ∗ ( a ∗ c ) = ( b ∗ a ) ∗ c = c b=b*e=b*(a*c)=(b*a)*c=c b = b ∗ e = b ∗ ( a ∗ c ) = ( b ∗ a ) ∗ c = c a a a
定義 5 - 3.1 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ S S S ∗ * ∗ S S S ∗ * ∗ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ 廣羣 。
定義 5 - 3.2 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ S S S ∗ * ∗ S S S ∗ * ∗ ∗ * ∗ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ 半羣 。
定理 5 - 3.1 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ B ⊆ S B\subseteq S B ⊆ S ∗ * ∗ B B B ⟨ B , ∗ ⟩ \langle B,* \rangle ⟨ B , ∗ ⟩ ⟨ B , ∗ ⟩ \langle B,* \rangle ⟨ B , ∗ ⟩ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ 子半羣 。
定理 5 - 3.2 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ S S S a ∈ S a\in S a ∈ S a ∗ a = a a*a=a a ∗ a = a
證明： S S S b b b j > i j>i j > i b i = b j b^i=b^j b i = b j p = j − i p=j-i p = j − i b i = b p ∗ b i b^i=b^p*b^i b i = b p ∗ b i b q = b p ∗ b q , q ⩾ i b^q=b^p*b^q,q\geqslant i b q = b p ∗ b q , q ⩾ i p ⩾ 1 p\geqslant1 p ⩾ 1 k ⩾ 1 k\geqslant1 k ⩾ 1 k p ⩾ i kp\geqslant i k p ⩾ i b k p = b p ∗ b k p = b p ∗ ( b p ∗ b k p ) = ⋯ = b k p ∗ b k p b^{kp}=b^p*b^{kp}=b^p*(b^p*b^{kp})=\cdots=b^{kp}*b^{kp} b k p = b p ∗ b k p = b p ∗ ( b p ∗ b k p ) = ⋯ = b k p ∗ b k p
定義 5 - 3.3 獨異點 。
定理 5 - 3.3 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ∗ \ast ∗
證明： 含幺元 e e e
定理 5 - 3.4 ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ a , b ∈ S a,b\in S a , b ∈ S a , b a,b a , b ( a − 1 ) − 1 = a (a^{-1})^{-1}=a ( a − 1 ) − 1 = a a ∗ b a*b a ∗ b ( a ∗ b ) − 1 = b − 1 ∗ a − 1 (a*b)^{-1}=b^{-1}*a^{-1} ( a ∗ b ) − 1 = b − 1 ∗ a − 1
定義 5 - 4.1 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ G G G ∗ * ∗ G G G ∗ * ∗ ∗ * ∗ e e e x ∈ G x\in G x ∈ G x − 1 x^{-1} x − 1 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ 羣 。
定義 5 - 4.2 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ G G G ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ 有限羣 ， G G G 階數 ，記爲 ∣ G ∣ \vert G \vert ∣ G ∣ G G G ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ 無限羣 。
定理 5 - 4.1
證明： 零元 θ \theta θ
定理 5 - 4.2 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ a , b ∈ G a,b\in G a , b ∈ G x ∈ G x\in G x ∈ G a ∗ x = b a*x=b a ∗ x = b
定理 5 - 4.3 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ a , b , c ∈ G a,b,c\in G a , b , c ∈ G a ∗ b = a ∗ c a*b=a*c a ∗ b = a ∗ c b ∗ a = c ∗ a b*a=c*a b ∗ a = c ∗ a b = c b=c b = c 消去律 ）。
定義 5 - 4.3 S S S S S S S S S S S S 置換 。
定理 5 - 4.4 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ G G G
證明： G G G G G G 定理 5 - 3.3 即證。
定義 5 - 4.4 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ a ∈ G a\in G a ∈ G a ∗ a = a a*a=a a ∗ a = a a a a 等冪元 。
定理 5 - 4.5 ⟨ A , ∗ ⟩ \langle A,* \rangle ⟨ A , ∗ ⟩ e e e
證明： e ≠ a = e ∗ a = ( a − 1 ∗ a ) ∗ a = a − 1 ∗ ( a ∗ a ) = a − 1 ∗ a = e e\ne a=e*a=(a^{-1}*a)*a=a^{-1}*(a*a)=a^{-1}*a=e e ̸ = a = e ∗ a = ( a − 1 ∗ a ) ∗ a = a − 1 ∗ ( a ∗ a ) = a − 1 ∗ a = e
定義 5 - 4.5 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ S S S G G G ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ 子羣 。
定理 5 - 4.6 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ e e e ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩
證明： e G ∗ x = x = e S ∗ x ⇔ e G = e S e_G*x=x=e_S*x\Leftrightarrow e_G=e_S e G ∗ x = x = e S ∗ x ⇔ e G = e S
定義 5 - 4.6 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ S = { e } S=\{e\} S = { e } S = G S=G S = G ⟨ S , ∗ ⟩ \langle S,* \rangle ⟨ S , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ 平凡子羣 。
定理 5 - 4.7 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ B B B G G G B B B ∗ * ∗ B B B ⟨ B , ∗ ⟩ \langle B,* \rangle ⟨ B , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩
證明： b ∈ B , b i = b j , i < j b\in B,b^i=b^j,i<j b ∈ B , b i = b j , i < j b i = b j − i ∗ b i b^i=b^{j-i}*b^i b i = b j − i ∗ b i b j − i b^{j-i} b j − i j − i > 1 j-i>1 j − i > 1 b j − i = b ∗ b j − i − 1 b^{j-i}=b*b^{j-i-1} b j − i = b ∗ b j − i − 1 b j − i − 1 b^{j-i-1} b j − i − 1 b b b j − i = 1 j-i=1 j − i = 1 b i = b ∗ b i b^i=b*b^i b i = b ∗ b i b b b
定理 5 - 4.8 ⟨ G , △ ⟩ \langle G,\triangle \rangle ⟨ G , △ ⟩ S S S G G G S S S a a a b b b a △ b − 1 ∈ S a\triangle b^{-1}\in S a △ b − 1 ∈ S ⟨ S , △ ⟩ \langle S,\triangle \rangle ⟨ S , △ ⟩ ⟨ G , △ ⟩ \langle G,\triangle \rangle ⟨ G , △ ⟩
證明： G G G S S S a △ a − 1 = e ∈ S a\triangle a^{-1}=e\in S a △ a − 1 = e ∈ S S S S a − 1 = e △ a − 1 ∈ S a^{-1}=e\triangle a^{-1}\in S a − 1 = e △ a − 1 ∈ S △ \triangle △ S S S a △ b = a △ ( b − 1 ) − 1 ∈ S a\triangle b=a\triangle(b^{-1})^{-1}\in S a △ b = a △ ( b − 1 ) − 1 ∈ S
定義 5 - 5.1 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ ∗ * ∗ 阿貝爾羣 ，或稱交換羣 。
定理 5 - 5.1 ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ a , b ∈ G a,b\in G a , b ∈ G ( a ∗ b ) ∗ ( a ∗ b ) = ( a ∗ a ) ∗ ( b ∗ b ) (a*b)*(a*b)=(a*a)*(b*b) ( a ∗ b ) ∗ ( a ∗ b ) = ( a ∗ a ) ∗ ( b ∗ b )
證明： a ∗ ( a ∗ b ) ∗ b = ( a ∗ a ) ∗ ( b ∗ b ) = ( a ∗ b ) ∗ ( a ∗ b ) = a ∗ ( b ∗ a ) ∗ b \begin{aligned}a*(a*b)*b&=(a*a)*(b*b)\\&=(a*b)*(a*b)\\&=a*(b*a)*b\end{aligned} a ∗ ( a ∗ b ) ∗ b = ( a ∗ a ) ∗ ( b ∗ b ) = ( a ∗ b ) ∗ ( a ∗ b ) = a ∗ ( b ∗ a ) ∗ b a ∗ b = b ∗ a a*b=b*a a ∗ b = b ∗ a ( a ∗ a ) ∗ ( b ∗ b ) = a ∗ ( a ∗ b ) ∗ b = a ∗ ( b ∗ a ) ∗ b = ( a ∗ b ) ∗ ( a ∗ b ) \begin{aligned}(a*a)*(b*b)&=a*(a*b)*b\\&=a*(b*a)*b\\&=(a*b)*(a*b)\end{aligned} ( a ∗ a ) ∗ ( b ∗ b ) = a ∗ ( a ∗ b ) ∗ b = a ∗ ( b ∗ a ) ∗ b = ( a ∗ b ) ∗ ( a ∗ b )
定義 5 - 5.2 ⟨ G , ∗ ⟩ \langle G,* \rangle ⟨ G , ∗ ⟩ G G G a a a G G G a a a 循環羣 ，元素 a a a G G G 生成元 。
定理 5 - 5.2
證明： x = a r , y = a s x=a^r,y=a^s x = a r , y = a s x ∗ y = a r ∗ a s = a r + s = a s + r = a s ∗ a r = y ∗ x x*y=a^r*a^s=a^{r+s}=a^{s+r}=a^s*a^r=y*x x ∗ y = a r ∗ a s = a r + s = a s + r = a s ∗ a r = y ∗ x
定理 5 - 5.3 ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ a ∈ G a\in G a ∈ G G G G n n n ∣ G ∣ = n \vert G\vert=n ∣ G ∣ = n a n = e a^n=e a n = e G = { a , a 2 , a 3 , ⋯  , a n − 1 , a n = e } G=\{a,a^2,a^3,\cdots,a^{n-1},a^n=e\} G = { a , a 2 , a 3 , ⋯ , a n − 1 , a n = e } e e e ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ n n n a n = e a^n=e a n = e n n n a a a
證明： m < n m<n m < n a m = e a^m=e a m = e ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ G G G a k a^k a k k = m q + r , q ∈ I , 0 ⩽ r < m k=mq+r,q\in I,0\leqslant r<m k = m q + r , q ∈ I , 0 ⩽ r < m G G G a r a^r a r G G G m m m ∣ G ∣ = n \vert G\vert=n ∣ G ∣ = n
定義 5 - 6.1 π 1 , π 2 ∈ S n \pi_1,\pi_2\in S_n π 1 , π 2 ∈ S n S n S_n S n ∘ \circ ∘ ⋄ \diamond ⋄ π 1 ∘ π 2 \pi_1\circ\pi_2 π 1 ∘ π 2 π 2 ⋄ π 1 \pi_2\diamond\pi_1 π 2 ⋄ π 1 S S S π 2 \pi_2 π 2 π 1 \pi_1 π 1 ∘ \circ ∘ ⋄ \diamond ⋄ 左複合 和右複合 。
定理 5 - 6.1 ⟨ S n , ∘ ⟩ \langle S_n,\circ\rangle ⟨ S n , ∘ ⟩ ∘ \circ ∘
證明： ∘ \circ ∘ S n S_n S n π 2 ( a ) = c , π 2 ( b ) = d , a ≠ b \pi_2(a)=c,\pi_2(b)=d,a\ne b π 2 ( a ) = c , π 2 ( b ) = d , a ̸ = b c ≠ d c\ne d c ̸ = d π 1 ( c ) = e , π 1 ( d ) = f \pi_1(c)=e,\pi_1(d)=f π 1 ( c ) = e , π 1 ( d ) = f e ≠ f e\ne f e ̸ = f π 1 ∘ π 2 ∈ S n \pi_1\circ\pi_2\in S_n π 1 ∘ π 2 ∈ S n ∘ \circ ∘ S n S_n S n π 3 ( x ) = y , π 2 ( y ) = z , π 1 ( z ) = w \pi_3(x)=y,\pi_2(y)=z,\pi_1(z)=w π 3 ( x ) = y , π 2 ( y ) = z , π 1 ( z ) = w ( π 1 ∘ π 2 ) ∘ π 3 ( x ) = ( π 1 ∘ π 2 ) y = π 1 ( z ) = w (\pi_1\circ\pi_2)\circ\pi_3(x)=(\pi_1\circ\pi_2)y=\pi_1(z)=w ( π 1 ∘ π 2 ) ∘ π 3 ( x ) = ( π 1 ∘ π 2 ) y = π 1 ( z ) = w π 1 ∘ ( π 2 ∘ π 3 ( x ) ) = π 1 ∘ ( π 2 ( y ) ) = π 1 ( z ) = w \pi_1\circ(\pi_2\circ\pi_3(x))=\pi_1\circ(\pi_2(y))=\pi_1(z)=w
π 1 ∘ ( π 2 ∘ π 3 ( x ) ) = π 1 ∘ ( π 2 ( y ) ) = π 1 ( z ) = w S n S_n S n π e \pi_e π e ∀ π x ∈ S n , π e ∘ π x = π x \forall \pi_x\in S_n,\pi_e\circ\pi_x=\pi_x ∀ π x ∈ S n , π e ∘ π x = π x π e \pi_e π e S n S_n S n ∀ π ∈ S , ∃ π − 1 ∈ S n \forall\pi\in S,\exist\pi^{-1}\in S_n ∀ π ∈ S , ∃ π − 1 ∈ S n π ( x ) = y → π − 1 ( y ) = x \pi(x)=y\rightarrow\pi^{-1}(y)=x π ( x ) = y → π − 1 ( y ) = x
定義 5 - 6.2 ⟨ S n , ∗ ⟩ \langle S_n,*\rangle ⟨ S n , ∗ ⟩ S S S 置換羣 。特別地，置換羣 ⟨ S n , ∘ ⟩ \langle S_n,\circ\rangle ⟨ S n , ∘ ⟩ S S S 對稱羣 。
定義 5 - 6.3 ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ S S S R = { ⟨ a , b ⟩ ∣ π ( a ) = b , π ∈ G } R=\{\langle a,b\rangle\vert\pi(a)=b,\pi\in G\} R = { ⟨ a , b ⟩ ∣ π ( a ) = b , π ∈ G } 由 ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ S S S 。
定理 5 - 6.2 ⟨ G , ∘ ⟩ \langle G,\circ\rangle ⟨ G , ∘ ⟩ S S S
定義 5 - 6.4 不變元 。
定理 5 - 6.3 （伯恩賽德定理）S S S ⟨ G , ∘ ⟩ \langle G,\circ\rangle ⟨ G , ∘ ⟩ S S S 1 ∣ G ∣ ∑ π ∈ G ψ ( π ) \frac{1}{\vert G\vert}\sum_{\pi\in G}\psi(\pi) ∣ G ∣ 1 π ∈ G ∑ ψ ( π ) ψ ( π ) \psi(\pi) ψ ( π ) π \pi π
定義 5 - 7.1 ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ A , B ∈ P ( G ) A,B\in\mathscr{P}(G) A , B ∈ P ( G ) A , B ≠ ∅ A,B\ne\varnothing A , B ̸ = ∅ A B = { a ∗ b ∣ a ∈ A , b ∈ B } AB=\{a*b\vert a\in A,b\in B\} A B = { a ∗ b ∣ a ∈ A , b ∈ B } A − 1 = { a − 1 ∣ a ∈ A } A^{-1}=\{a^{-1}\vert a\in A\} A − 1 = { a − 1 ∣ a ∈ A } A , B A,B A , B 積 和 A A A 逆 。
定義 5 - 7.2 ⟨ H , ∗ ⟩ \langle H,*\rangle ⟨ H , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ a ∈ G a\in G a ∈ G { a } H \{a\}H { a } H H { a } H\{a\} H { a } a a a H H H G G G 左（右）陪集 ，簡稱爲 H H H a a a a H aH a H H a Ha H a a a a a H aH a H H a Ha H a 代表元素 。
定理 5 - 7.1 （拉格朗日定理）⟨ H , ∗ ⟩ \langle H,*\rangle ⟨ H , ∗ ⟩ ⟨ G , ∗ ⟩ \langle G,*\rangle ⟨ G , ∗ ⟩ R = { ⟨ a , b ⟩ ∣ a ∈ G , b ∈ G , a − 1 ∗ b ∈ H } R=\{\langle a,b\rangle\vert a\in G,b\in G,a^{-1}*b\in H\} R = { ⟨ a , b ⟩ ∣ a ∈ G , b ∈ G , a − 1 ∗ b ∈ H } G G G a ∈ G a\in G a ∈ G [ a ] R = { x ∣ x ∈ G , ⟨ a , x ⟩ ∈ R } [a]_R=\{x\vert x\in G,\langle a,x\rangle\in R\} [ a ] R = { x ∣ x ∈ G , ⟨ a , x ⟩ ∈ R } [ a ] R = a H [a]_R=aH [ a ] R = a H G G G ∣ G ∣ = n \vert G\vert=n ∣ G ∣ = n ∣ H ∣ = m \vert H\vert=m ∣ H ∣ = m m ∣ n m\mid n m ∣ n
定義 5 - 8.1 ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ ★ \bigstar ★ ∗ * ∗ A A A B B B n n n f f f A A A B B B a 1 , a 2 ∈ A a_1,a_2\in A a 1 , a 2 ∈ A f ( a 1 ★ a 2 ) = f ( a 1 ) ∗ f ( a 2 ) f(a_1\bigstar a_2)=f(a_1)*f(a_2) f ( a 1 ★ a 2 ) = f ( a 1 ) ∗ f ( a 2 ) f f f ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ 同態映射 ，稱 ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ A ∼ B A\sim B A ∼ B ⟨ f ( A ) , ∗ ⟩ \langle f(A),*\rangle ⟨ f ( A ) , ∗ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ 同態象 。其中 f ( A ) = { x ∣ x = f ( a ) , a ∈ A } ⊆ B f(A)=\{x\vert x=f(a),a\in A\}\subseteq B f ( A ) = { x ∣ x = f ( a ) , a ∈ A } ⊆ B
定義 5 - 8.2 f f f ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ f f f A A A B B B f f f 滿同態 ；如果 f f f A A A B B B f f f 單一同態 ；如果 f f f A A A B B B f f f 同構映射 ，並稱 ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ 同構 的，記作 A ≅ B A\cong B A ≅ B
定義 5 - 8.3 ⟨ A , ⋆ ⟩ \langle A,\star\rangle ⟨ A , ⋆ ⟩ f f f ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ f f f 自同態 。如果 g g g ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ f f f 自同構 。
定理 5 - 8.1 G G G G G G
定理 5 - 8.2 f f f ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,*\rangle ⟨ B , ∗ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar\rangle ⟨ A , ★ ⟩ f f f ⟨ f ( A ) , ∗ ⟩ \langle f(A),* \rangle ⟨ f ( A ) , ∗ ⟩
定義 5 - 8.4 f f f ⟨ G , ★ ⟩ \langle G,\bigstar\rangle ⟨ G , ★ ⟩ ⟨ G ′ , ∗ ⟩ \langle G^\prime,*\rangle ⟨ G ′ , ∗ ⟩ e ′ e^\prime e ′ G ′ G^\prime G ′ Ker ( f ) = { x ∣ x ∈ G , f ( x ) = e ′ } \text{Ker}(f)=\{x\vert x\in G,f(x)=e^\prime\} Ker ( f ) = { x ∣ x ∈ G , f ( x ) = e ′ } Ker ( f ) \text{Ker}(f) Ker ( f ) f f f 核 ，簡稱 f f f 同態核 。
定理 5 - 8.3 f f f ⟨ G , ★ ⟩ \langle G,\bigstar\rangle ⟨ G , ★ ⟩ ⟨ G ′ , ∗ ⟩ \langle G^\prime,*\rangle ⟨ G ′ , ∗ ⟩ f f f K K K G G G
證明： k 1 , k 2 ∈ K k_1,k_2\in K k 1 , k 2 ∈ K f ( k 1 ★ k 2 ) = f ( k 1 ) ∗ f ( k 2 ) = e ′ ∗ e ′ = e ′ f(k_1\bigstar k _2)=f(k_1)*f(k_2)=e^\prime*e^\prime=e^\prime f ( k 1 ★ k 2 ) = f ( k 1 ) ∗ f ( k 2 ) = e ′ ∗ e ′ = e ′ k 1 ★ k 2 ∈ K k_1\bigstar k_2\in K k 1 ★ k 2 ∈ K f ( k − 1 ) = f ( k ) − 1 = e ′ − 1 = e ′ f(k^{-1})=f^{}(k)^{-1}=e^{\prime-1}=e^\prime f ( k − 1 ) = f ( k ) − 1 = e ′ − 1 = e ′ k − 1 ∈ K k^{-1}\in K k − 1 ∈ K ⟨ K , ★ ⟩ \langle K,\bigstar \rangle ⟨ K , ★ ⟩ ⟨ G , ★ ⟩ \langle G,\bigstar \rangle ⟨ G , ★ ⟩
定義 5 - 8.5 ⟨ A , ★ ⟩ \langle A,\bigstar \rangle ⟨ A , ★ ⟩ R R R A A A ⟨ a 1 , a 2 ⟩ , ⟨ b 1 , b 2 ⟩ ∈ R \langle a_1,a_2 \rangle,\langle b_1,b_2\rangle\in R ⟨ a 1 , a 2 ⟩ , ⟨ b 1 , b 2 ⟩ ∈ R ⟨ a 1 ★ b 1 , a 2 ★ b 2 ⟩ ∈ R \langle a_1\bigstar b_1,a_2\bigstar b_2\rangle\in R ⟨ a 1 ★ b 1 , a 2 ★ b 2 ⟩ ∈ R R R R A A A ★ \bigstar ★ 同餘關係 。由這個同餘關係將 A A A 同餘類 。
定理 5 - 8.4 ⟨ A , ★ ⟩ \langle A,\bigstar \rangle ⟨ A , ★ ⟩ R R R A A A B = { A 1 , A 2 , ⋯  , A r } B=\{A_1,A_2,\cdots,A_r\} B = { A 1 , A 2 , ⋯ , A r } R R R A A A ⟨ B , ∗ ⟩ \langle B,* \rangle ⟨ B , ∗ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar \rangle ⟨ A , ★ ⟩
定理 5 - 8.5 f f f ⟨ A , ★ ⟩ \langle A,\bigstar \rangle ⟨ A , ★ ⟩ ⟨ B , ∗ ⟩ \langle B,* \rangle ⟨ B , ∗ ⟩ A A A R R R ⟨ a , b ⟩ ∈ R \langle a,b \rangle \in R ⟨ a , b ⟩ ∈ R f ( a ) = f ( b ) f(a)=f(b) f ( a ) = f ( b ) R R R A A A
形象地說，一個代數系統的同態象可以看作是當抽去該系統中某些元素的次要特徵的情況下，對該系統的一種粗糙描述。如果我們把屬於同一個同餘類的元素看作是沒有區別的，那麼原系統的性態可以用同餘類之間的相互關係來描述。
定義 5 - 9.1 ⟨ A , ★ , ∗ ⟩ \langle A,\bigstar,*\rangle ⟨ A , ★ , ∗ ⟩ ⟨ A , ★ ⟩ \langle A,\bigstar \rangle ⟨ A , ★ ⟩ ⟨ A , ∗ ⟩ \langle A,* \rangle ⟨ A , ∗ ⟩ ∗ * ∗ ★ \bigstar ★ ⟨ A , ★ , ∗ ⟩ \langle A,\bigstar,*\rangle ⟨ A , ★ , ∗ ⟩ 環 。
定理 5 - 9.1 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ a , b , c ∈ A a,b,c\in A a , b , c ∈ A a ⋅ θ = θ ⋅ a = θ a\cdot\theta=\theta\cdot a=\theta a ⋅ θ = θ ⋅ a = θ a ⋅ ( − b ) = ( − a ) ⋅ b = − ( a ⋅ b ) a\cdot(-b)=(-a)\cdot b=-(a\cdot b) a ⋅ ( − b ) = ( − a ) ⋅ b = − ( a ⋅ b ) ( − a ) ⋅ ( − b ) = a ⋅ b (-a)\cdot(-b)=a\cdot b ( − a ) ⋅ ( − b ) = a ⋅ b a ⋅ ( b − c ) = a ⋅ b − a ⋅ c a\cdot(b-c)=a\cdot b-a\cdot c a ⋅ ( b − c ) = a ⋅ b − a ⋅ c ( b − c ) ⋅ a = b ⋅ a − c ⋅ a (b-c)\cdot a=b\cdot a-c\cdot a ( b − c ) ⋅ a = b ⋅ a − c ⋅ a θ \theta θ − a -a − a a a a a + ( − b ) a+(-b) a + ( − b ) a − b a-b a − b
證明： θ ⋅ a = ( θ + θ ) ⋅ a = θ ⋅ a + θ ⋅ a \theta\cdot a=(\theta+\theta)\cdot a=\theta\cdot a+\theta\cdot a θ ⋅ a = ( θ + θ ) ⋅ a = θ ⋅ a + θ ⋅ a θ ⋅ a = θ \theta\cdot a=\theta θ ⋅ a = θ a ⋅ θ = θ a\cdot\theta=\theta a ⋅ θ = θ a ⋅ b + a ⋅ ( − b ) = a ⋅ [ b + ( − b ) ] = a ⋅ θ = θ a\cdot b+a\cdot(-b)=a\cdot[b+(-b)]=a\cdot\theta=\theta a ⋅ b + a ⋅ ( − b ) = a ⋅ [ b + ( − b ) ] = a ⋅ θ = θ a ⋅ ( − b ) = − ( a ⋅ b ) a\cdot(-b)=-(a\cdot b) a ⋅ ( − b ) = − ( a ⋅ b ) ( − a ) ⋅ b = − ( a ⋅ b ) (-a)\cdot b=-(a\cdot b) ( − a ) ⋅ b = − ( a ⋅ b )
定義 5 - 9.2 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ A , ⋅ ⟩ \langle A,\cdot\rangle ⟨ A , ⋅ ⟩ ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ 交換環 。如果 ⟨ A , ⋅ ⟩ \langle A,\cdot\rangle ⟨ A , ⋅ ⟩ ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ 含幺環 。
定義 5 - 9.3 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ A , + ⟩ \langle A,+\rangle ⟨ A , + ⟩ ⟨ A , ⋅ ⟩ \langle A,\cdot\rangle ⟨ A , ⋅ ⟩ a , b ∈ A , a ≠ θ , b ≠ θ a,b\in A,a\ne\theta,b\ne\theta a , b ∈ A , a ̸ = θ , b ̸ = θ a ⋅ b ≠ θ a\cdot b\ne\theta a ⋅ b ̸ = θ ⋅ \cdot ⋅ + + + ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ 整環 。
定理 5 - 9.2 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ c ≠ θ c\ne\theta c ̸ = θ c ⋅ a = c ⋅ b c\cdot a=c\cdot b c ⋅ a = c ⋅ b a = b a=b a = b
定義 5 - 9.4 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ A , + ⟩ \langle A,+\rangle ⟨ A , + ⟩ ⟨ A − { θ } , ⋅ ⟩ \langle A-\{\theta\},\cdot\rangle ⟨ A − { θ } , ⋅ ⟩ ⋅ \cdot ⋅ + + + ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ 域 。
定理 5 - 9.3
定理 5 - 9.4
定義 5 - 9.5 ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ B , ⊕ , ⊙ ⟩ \langle B,\oplus,\odot \rangle ⟨ B , ⊕ , ⊙ ⟩ A A A B B B f f f a , b ∈ A a,b\in A a , b ∈ A f ( a + b ) = f ( a ) ⊕ f ( b ) f(a+b)=f(a)\oplus f(b) f ( a + b ) = f ( a ) ⊕ f ( b ) f ( a ⋅ b ) = f ( a ) ⊙ f ( b ) f(a\cdot b) = f(a)\odot f(b) f ( a ⋅ b ) = f ( a ) ⊙ f ( b ) f f f ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ B , ⊕ , ⊙ ⟩ \langle B,\oplus,\odot \rangle ⟨ B , ⊕ , ⊙ ⟩ 同態映射 ，並稱 ⟨ f ( A ) , ⊕ , ⊙ ⟩ \langle f(A),\oplus,\odot\rangle ⟨ f ( A ) , ⊕ , ⊙ ⟩ ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot \rangle ⟨ A , + , ⋅ ⟩ 同態象 。
定理 5 - 9.5
證明： ⟨ A , + , ⋅ ⟩ \langle A,+,\cdot\rangle ⟨ A , + , ⋅ ⟩ ⟨ B , ⊕ , ⊙ ⟩ \langle B,\oplus,\odot \rangle ⟨ B , ⊕ , ⊙ ⟩ f f f ⟨ A , + ⟩ \langle A,+\rangle ⟨ A , + ⟩ ⟨ B , ⊕ ⟩ \langle B,\oplus\rangle ⟨ B , ⊕ ⟩ ⟨ A , ⋅ ⟩ \langle A,\cdot\rangle ⟨ A , ⋅ ⟩ ⟨ B , ⊙ ⟩ \langle B,\odot\rangle ⟨ B , ⊙ ⟩ b 1 , b 2 , b 3 ∈ B b_1,b_2,b_3\in B b 1 , b 2 , b 3 ∈ B a 1 , a 2 , a 3 a_1,a_2,a_3 a 1 , a 2 , a 3 f ( a i ) = b i , ( i = 1 , 2 , 3 ) f(a_i)=b_i,(i=1,2,3) f ( a i ) = b i , ( i = 1 , 2 , 3 ) b 1 ⊙ ( b 2 ⊕ b 3 ) = f ( a 1 ) ⊙ ( f ( a 2 ) ⊕ f ( a 3 ) ) = f ( a 1 ) ⊙ f ( a 2 + a 3 ) = f ( a 1 ⋅ ( a 2 + a 3 ) ) = f ( ( a 1 ⋅ a 2 ) + ( a 1 ⋅ a 3 ) ) = f ( a 1 ⋅ a 2 ) ⊕ f ( a 1 ⋅ a 3 ) = ( f ( a 1 ) ⊙ f ( a 2 ) ) ⊕ ( f ( a 1 ) ⊙ f ( a 3 ) ) = ( b 1 ⊙ b 2 ) ⊕ ( b 1 ⊙ b 3 ) \begin{aligned}b_1\odot(b_2\oplus b_3)&=f(a_1)\odot(f(a_2)\oplus f(a_3))\\&=f(a_1)\odot f(a_2+a_3)\\&=f(a_1\cdot(a_2+a_3))\\&=f((a_1\cdot a_2)+(a_1\cdot a_3))\\&=f(a_1\cdot a_2)\oplus f(a_1\cdot a_3)\\&=(f(a_1)\odot f(a_2))\oplus(f(a_1)\odot f(a_3))\\&=(b_1\odot b_2)\oplus(b_1\odot b_3)\end{aligned} b 1 ⊙ ( b 2 ⊕ b 3 ) = f ( a 1 ) ⊙ ( f ( a 2 ) ⊕ f ( a 3 ) ) = f ( a 1 ) ⊙ f ( a 2 + a 3 ) = f ( a 1 ⋅ ( a 2 + a 3 ) ) = f ( ( a 1 ⋅ a 2 ) + ( a 1 ⋅ a 3 ) ) = f ( a 1 ⋅ a 2 ) ⊕ f ( a 1 ⋅ a 3 ) = ( f ( a 1 ) ⊙ f ( a 2 ) ) ⊕ ( f ( a 1 ) ⊙ f ( a 3 ) ) = ( b 1 ⊙ b 2 ) ⊕ ( b 1 ⊙ b 3 )
同理可證 ( b 2 ⊕ b 3 ) ⊙ b 1 = ( b 2 ⊙ b 1 ) ⊕ ( b 2 ⊙ b 1 ) (b_2\oplus b_3)\odot b_1=(b_2\odot b_1)\oplus(b_2\odot b_1) ( b 2 ⊕ b 3 ) ⊙ b 1 = ( b 2 ⊙ b 1 ) ⊕ ( b 2 ⊙ b 1 ) ⟨ B , ⊕ , ⊙ ⟩ \langle B,\oplus,\odot \rangle ⟨ B , ⊕ , ⊙ ⟩
定義 6 - 1.1 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ A A A ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ 格 。
定義 6 - 1.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ A A A ∨ \vee ∨ ∧ \wedge ∧ a , b ∈ A a,b\in A a , b ∈ A a ∨ b a\vee b a ∨ b a a a b b b a ∧ b a\wedge b a ∧ b a a a b b b ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ 由格 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ 。二元運算 ∨ \vee ∨ ∧ \wedge ∧ 並運算 和交運算 。
定義 6 - 1.3 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ B ⊆ A B\subseteq A B ⊆ A B ≠ ∅ B\ne\varnothing B ̸ = ∅ A A A ∨ \vee ∨ ∧ \wedge ∧ ⟨ B , ≼ ⟩ \langle B,\preccurlyeq\rangle ⟨ B , ≼ ⟩ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ 子格 。
定理 6 - 1.1 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ a , b ∈ A a,b\in A a , b ∈ A a ≼ a ∨ b a\preccurlyeq a\vee b a ≼ a ∨ b b ≼ a ∨ b b\preccurlyeq a\vee b b ≼ a ∨ b a ∧ b ≼ a a\wedge b \preccurlyeq a a ∧ b ≼ a a ∧ b ≼ b a\wedge b \preccurlyeq b a ∧ b ≼ b
證明：
定理 6 - 1.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ a , b , c , d ∈ A a,b,c,d\in A a , b , c , d ∈ A a ≼ b a\preccurlyeq b a ≼ b c ≼ d c\preccurlyeq d c ≼ d a ∨ c ≼ b ∨ d a\vee c \preccurlyeq b \vee d a ∨ c ≼ b ∨ d a ∧ c ≼ b ∧ d a\wedge c \preccurlyeq b \wedge d a ∧ c ≼ b ∧ d
證明： a ≼ b ≼ b ∨ d , c ≼ d ≼ b ∨ d a \preccurlyeq b \preccurlyeq b\vee d,c \preccurlyeq d \preccurlyeq b\vee d a ≼ b ≼ b ∨ d , c ≼ d ≼ b ∨ d a ∨ c ≼ b ∨ d a\vee c \preccurlyeq b\vee d a ∨ c ≼ b ∨ d a ∧ c ≼ b ∧ d a\wedge c \preccurlyeq b\wedge d a ∧ c ≼ b ∧ d
定理 6 - 1.3 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq\rangle ⟨ A , ≼ ⟩ ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ a , b , c , d ∈ A a,b,c,d\in A a , b , c , d ∈ A a ∨ b = b ∨ a a\vee b=b\vee a a ∨ b = b ∨ a a ∨ ( b ∨ c ) = ( a ∨ b ) ∨ c a\vee(b\vee c)=(a\vee b)\vee c a ∨ ( b ∨ c ) = ( a ∨ b ) ∨ c a ∨ a = a a\vee a=a a ∨ a = a a ∨ ( a ∧ b ) = a a\vee(a\wedge b)=a a ∨ ( a ∧ b ) = a
證明：
（2）b ≼ b ∨ c ≼ a ∨ ( b ∨ c ) a ≼ a ∨ ( b ∨ c ) ⇒ ( a ∨ b ) ≼ a ∨ ( b ∨ c ) \begin{aligned}&b\preccurlyeq b\vee c \preccurlyeq a \vee (b \vee c)\\&a\preccurlyeq a \vee (b \vee c) \\\Rightarrow &(a\vee b)\preccurlyeq a\vee(b\vee c)\end{aligned} ⇒ b ≼ b ∨ c ≼ a ∨ ( b ∨ c ) a ≼ a ∨ ( b ∨ c ) ( a ∨ b ) ≼ a ∨ ( b ∨ c ) c ≼ b ∨ c ≼ a ∨ ( b ∨ c ) c\preccurlyeq b\vee c \preccurlyeq a\vee (b\vee c) c ≼ b ∨ c ≼ a ∨ ( b ∨ c ) ( a ∨ b ) ∨ c ≼ a ∨ ( b ∨ c ) (a\vee b)\vee c\preccurlyeq a\vee (b \vee c) ( a ∨ b ) ∨ c ≼ a ∨ ( b ∨ c ) a ∨ ( b ∨ c ) ≼ ( a ∨ b ) ∨ c a\vee (b \vee c)\preccurlyeq (a\vee b)\vee c a ∨ ( b ∨ c ) ≼ ( a ∨ b ) ∨ c ( a ∨ b ) ∨ c = a ∨ ( b ∨ c ) (a\vee b)\vee c= a\vee (b \vee c) ( a ∨ b ) ∨ c = a ∨ ( b ∨ c )
（3）a ≼ a ∨ a , a ≼ a ⇒ a ∨ a ≼ a ⇒ a = a ∨ a \begin{aligned}&a\preccurlyeq a\vee a,\\&a\preccurlyeq a \Rightarrow a\vee a \preccurlyeq a\\\Rightarrow &a=a\vee a\end{aligned} ⇒ a ≼ a ∨ a , a ≼ a ⇒ a ∨ a ≼ a a = a ∨ a a ∧ a = a a\wedge a=a a ∧ a = a
（4）略
引理 6 - 1.1 ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ ∨ , ∧ \vee,\wedge ∨ , ∧ ∨ \vee ∨ ∧ \wedge ∧
證明： a ∨ ( a ∧ b ) = a a\vee(a\wedge b)=a a ∨ ( a ∧ b ) = a a ∧ ( a ∨ b ) a\wedge(a\vee b) a ∧ ( a ∨ b ) b b b a ∨ b a\vee b a ∨ b a ∨ ( a ∧ ( a ∨ b ) ) = a a\vee(a\wedge(a\vee b))=a a ∨ ( a ∧ ( a ∨ b ) ) = a a ∨ a = a a\vee a=a a ∨ a = a a ∧ a = a a\wedge a=a a ∧ a = a
定理 6 - 1.4 ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ ∨ \vee ∨ ∧ \wedge ∧ A A A ≼ \preccurlyeq ≼ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩
定理 6 - 1.5 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a , b , c ∈ A a,b,c\in A a , b , c ∈ A a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ ( a ∨ c ) a\vee(b\wedge c)\preccurlyeq(a\vee b)\wedge(a\vee c) a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ ( a ∨ c ) ( a ∧ b ) ∨ ( a ∧ c ) ≼ a ∧ ( b ∨ c ) (a\wedge b)\vee(a\wedge c)\preccurlyeq a\wedge(b\vee c) ( a ∧ b ) ∨ ( a ∧ c ) ≼ a ∧ ( b ∨ c )
定理 6 - 1.6 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a , b ∈ A a,b\in A a , b ∈ A a ≼ b ⇔ a ∧ b = a ⇔ a ∨ b = b a \preccurlyeq b \Leftrightarrow a \wedge b = a \Leftrightarrow a \vee b = b a ≼ b ⇔ a ∧ b = a ⇔ a ∨ b = b
定理 6 - 1.7 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a , b ∈ A a,b\in A a , b ∈ A a ≼ c ⇔ a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ c a \preccurlyeq c \Leftrightarrow a \vee (b \wedge c) \preccurlyeq (a \vee b) \wedge c a ≼ c ⇔ a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ c
定義 6 - 1.4 ⟨ A 1 , ≼ 1 ⟩ \langle A_1,\preccurlyeq_1 \rangle ⟨ A 1 , ≼ 1 ⟩ ⟨ A 2 , ≼ 2 ⟩ \langle A_2,\preccurlyeq_2 \rangle ⟨ A 2 , ≼ 2 ⟩ ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ \langle A_1,\vee_1,\wedge_1\rangle ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ \langle A_2,\vee_2,\wedge_2\rangle ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ A 1 A_1 A 1 A 2 A_2 A 2 f f f a , b ∈ A 1 a,b\in A_1 a , b ∈ A 1 f ( a ∨ 1 b ) = f ( a ) ∨ 2 f ( b ) f(a \vee_1 b)=f(a) \vee_2 f(b) f ( a ∨ 1 b ) = f ( a ) ∨ 2 f ( b ) f ( a ∧ 1 b ) = f ( a ) ∧ 2 f ( b ) f(a \wedge_1 b)=f(a) \wedge_2 f(b) f ( a ∧ 1 b ) = f ( a ) ∧ 2 f ( b ) f f f ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ \langle A_1,\vee_1,\wedge_1\rangle ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ \langle A_2,\vee_2,\wedge_2\rangle ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ 格同態象 。此外，當 f f f f f f ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ \langle A_1,\vee_1,\wedge_1\rangle ⟨ A 1 , ∨ 1 , ∧ 1 ⟩ ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ \langle A_2,\vee_2,\wedge_2\rangle ⟨ A 2 , ∨ 2 , ∧ 2 ⟩ 格同構 ，亦稱 ⟨ A 1 , ≼ 1 ⟩ \langle A_1,\preccurlyeq_1 \rangle ⟨ A 1 , ≼ 1 ⟩ ⟨ A 2 , ≼ 2 ⟩ \langle A_2,\preccurlyeq_2 \rangle ⟨ A 2 , ≼ 2 ⟩ 同構 的。
定理 6 - 1.8 f f f ⟨ A 1 , ≼ 1 ⟩ \langle A_1,\preccurlyeq_1 \rangle ⟨ A 1 , ≼ 1 ⟩ ⟨ A 2 , ≼ 2 ⟩ \langle A_2,\preccurlyeq_2 \rangle ⟨ A 2 , ≼ 2 ⟩ x , y ∈ A 1 x,y\in A_1 x , y ∈ A 1 x ≼ 1 y x\preccurlyeq_1 y x ≼ 1 y f ( x ) ≼ 2 f ( y ) f(x)\preccurlyeq_2 f(y) f ( x ) ≼ 2 f ( y )
定理 6 - 1.9 ⟨ A 1 , ≼ 1 ⟩ \langle A_1,\preccurlyeq_1 \rangle ⟨ A 1 , ≼ 1 ⟩ ⟨ A 2 , ≼ 2 ⟩ \langle A_2,\preccurlyeq_2 \rangle ⟨ A 2 , ≼ 2 ⟩ f f f A 1 A_1 A 1 A 2 A_2 A 2 f f f ⟨ A 1 , ≼ 1 ⟩ \langle A_1,\preccurlyeq_1 \rangle ⟨ A 1 , ≼ 1 ⟩ ⟨ A 2 , ≼ 2 ⟩ \langle A_2,\preccurlyeq_2 \rangle ⟨ A 2 , ≼ 2 ⟩ a , b ∈ A a,b\in A a , b ∈ A a ≼ 1 b ⇔ f ( a ) ≼ 2 f ( b ) a\preccurlyeq_1 b \Leftrightarrow f(a) \preccurlyeq_2 f(b) a ≼ 1 b ⇔ f ( a ) ≼ 2 f ( b )
定義 6 - 2.1 ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ ∧ \wedge ∧ ∨ \vee ∨ A A A ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 分配格 。
定理 6 - 2.1 ∧ \wedge ∧ ∨ \vee ∨ ∨ \vee ∨ ∧ \wedge ∧
定理 6 - 2.2
定理 6 - 2.3 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a , b , c ∈ A a,b,c\in A a , b , c ∈ A a ∧ b = a ∧ c a\wedge b = a\wedge c a ∧ b = a ∧ c a ∨ b = a ∨ c a\vee b = a \vee c a ∨ b = a ∨ c b = c b=c b = c
定義 6 - 2.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ ⟨ A , ∨ , ∧ ⟩ \langle A,\vee,\wedge\rangle ⟨ A , ∨ , ∧ ⟩ a , b , c ∈ A a,b,c \in A a , b , c ∈ A b ≼ a b \preccurlyeq a b ≼ a a ∧ ( b ∨ c ) = b ∨ ( a ∧ c ) a \wedge (b \vee c) = b \vee (a \wedge c) a ∧ ( b ∨ c ) = b ∨ ( a ∧ c ) ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 模格 。
定理 6 - 2.4 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ A A A u , v , w u,v,w u , v , w v ≺ u v \prec u v ≺ u u ∨ w = v ∨ w , u ∧ w = v ∧ w u \vee w = v \vee w, u \wedge w = v \wedge w u ∨ w = v ∨ w , u ∧ w = v ∧ w
定理 6 - 2.5 a , b , c a,b,c a , b , c a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ ( a ∨ c ) a \vee (b \wedge c) \preccurlyeq (a \vee b) \wedge (a \vee c) a ∨ ( b ∧ c ) ≼ ( a ∨ b ) ∧ ( a ∨ c ) ( a ∧ b ) ∨ ( a ∧ c ) ≼ a ∧ ( b ∨ c ) (a \wedge b) \vee (a \wedge c) \preccurlyeq a \wedge (b \vee c) ( a ∧ b ) ∨ ( a ∧ c ) ≼ a ∧ ( b ∨ c ) ( a ∧ b ) ∨ ( b ∧ c ) ∨ ( c ∧ a ) ≼ ( a ∨ b ) ∧ ( b ∨ c ) ∧ ( c ∨ a ) (a \wedge b) \vee (b \wedge c) \vee (c \wedge a) \preccurlyeq (a \vee b) \wedge (b \vee c) \wedge (c \vee a) ( a ∧ b ) ∨ ( b ∧ c ) ∨ ( c ∧ a ) ≼ ( a ∨ b ) ∧ ( b ∨ c ) ∧ ( c ∨ a ) a , b , c a,b,c a , b , c ≼ \preccurlyeq ≼ = = = ≼ \preccurlyeq ≼ = = =
定理 6 - 2.6
定義 6 - 3.1 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a ∈ A a\in A a ∈ A x ∈ A x\in A x ∈ A a ≼ x a\preccurlyeq x a ≼ x a a a ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 全下界 ，記格的全下界爲 0 0 0
定理 6 - 3.1
定義 6 - 3.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ b ∈ A b\in A b ∈ A x ∈ A x\in A x ∈ A x ≼ b x\preccurlyeq b x ≼ b b b b ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 全上界 ，記格的全上界爲 1 1 1
定理 6 - 3.2
定義 6 - 3.3 有界格 。
定理 6 - 3.3 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ a ∈ A a\in A a ∈ A a ∨ 1 = 1 , a ∧ 1 = a a\vee1=1,a\wedge1=a a ∨ 1 = 1 , a ∧ 1 = a a ∨ 0 = a , a ∧ 0 = 0 a\vee0=a,a\wedge0=0 a ∨ 0 = a , a ∧ 0 = 0
定義 6 - 3.4 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ A A A a a a b ∈ A b\in A b ∈ A a ∨ b = 1 a\vee b=1 a ∨ b = 1 a ∧ b = 0 a\wedge b=0 a ∧ b = 0 b b b a a a 補元 。
定義 6 - 3.5 有補格 。
定理 6 - 3.4
證明： a a a b b b c c c a ∨ b = 1 , a ∧ b = 0 a \vee b =1,a \wedge b=0 a ∨ b = 1 , a ∧ b = 0 a ∨ c = 1 , a ∧ c = 0 a \vee c = 1,a \wedge c =0 a ∨ c = 1 , a ∧ c = 0 定理 6 - 2.3 即得 b = c b=c b = c
定義 6 - 3.6 有補分配格 。我們把有補分配格中任一元素 a a a a ˉ \bar{a} a ˉ
定義 6 - 4.1 布爾格 。
定義 6 - 4.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ 布爾代數 。
定理 6 - 4.1 a , b a,b a , b ( a ˉ ) ‾ = a \overline{(\bar{a})}=a ( a ˉ ) = a a ∨ b ‾ = a ˉ ∧ b ˉ \overline{a \vee b} = \bar{a} \wedge \bar{b} a ∨ b = a ˉ ∧ b ˉ a ∧ b ‾ = a ˉ ∨ b ˉ \overline{a \wedge b} = \bar{a} \vee \bar{b} a ∧ b = a ˉ ∨ b ˉ
定義 6 - 4.3 有限布爾代數 。
定義 6 - 4.4 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ ⟨ B , ∨ , ∧ , ˉ ⟩ \langle B,\vee,\wedge,\bar{\quad} \rangle ⟨ B , ∨ , ∧ , ˉ ⟩ A A A B B B f f f a , b ∈ A a,b \in A a , b ∈ A f ( a ∨ b ) = f ( a ) ∨ f ( b ) f(a \vee b) = f(a) \vee f(b) f ( a ∨ b ) = f ( a ) ∨ f ( b ) f ( a ∧ b ) = f ( a ) ∧ f ( b ) f(a\wedge b)=f(a)\wedge f(b) f ( a ∧ b ) = f ( a ) ∧ f ( b ) f ( a ˉ ) = f ( a ) ‾ f(\bar{a})=\overline{f(a)} f ( a ˉ ) = f ( a ) ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ ⟨ B , ∨ , ∧ , ˉ ⟩ \langle B,\vee,\wedge,\bar{\quad} \rangle ⟨ B , ∨ , ∧ , ˉ ⟩ 同構 。
定義 6 - 4.5 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 0 0 0 a a a 0 0 0 a a a 原子 。
定理 6 - 4.2 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ 0 0 0 b b b 0 0 0 a a a a ≼ b a \preccurlyeq b a ≼ b
引理 6 - 4.1 b ∧ c ˉ = 0 b \wedge \bar{c} = 0 b ∧ c ˉ = 0 b ≼ c b \preccurlyeq c b ≼ c
引理 6 - 4.2 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ b b b A A A a 1 , a 2 , ⋯  , a k a_1,a_2,\cdots, a_k a 1 , a 2 , ⋯ , a k A A A a j ≼ b a_j \preccurlyeq b a j ≼ b j = 1 , 2 , ⋯  , k j=1,2,\cdots,k j = 1 , 2 , ⋯ , k b = a 1 ∨ a 2 ∨ ⋯ ∨ a k b = a_1 \vee a_2 \vee \cdots \vee a_k b = a 1 ∨ a 2 ∨ ⋯ ∨ a k
引理 6 - 4.3 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ b ∈ A b \in A b ∈ A b ≠ 0 b \ne 0 b ̸ = 0 a 1 , a 2 , ⋯  , a k a_1,a_2,\cdots,a_k a 1 , a 2 , ⋯ , a k a i ≼ b a_i \preccurlyeq b a i ≼ b i = 1 , 2 , ⋯  , k i=1,2,\cdots,k i = 1 , 2 , ⋯ , k A A A b = a 1 ∨ a 2 ∨ ⋯ ∨ a k b = a_1 \vee a_2 \vee \cdots \vee a_k b = a 1 ∨ a 2 ∨ ⋯ ∨ a k b b b
引理 6 - 4.4 ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ A A A a a a b b b a ≼ b a\preccurlyeq b a ≼ b a ≼ b ˉ a \preccurlyeq \bar{b} a ≼ b ˉ
定理 6 - 4.3 （Stone 表示定理）⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ S S S ⟨ A , ≼ ⟩ \langle A,\preccurlyeq \rangle ⟨ A , ≼ ⟩ ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ ⟨ P ( S ) , ∪ , ∩ , ∼ ⟩ \langle \mathscr{P}(S),\cup,\cap,\sim \rangle ⟨ P ( S ) , ∪ , ∩ , ∼ ⟩
定義 6 - 5.1 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ 布爾表達式 如下：A A A e 1 e_1 e 1 e 2 e_2 e 2 e ˉ 1 , ( e 1 ∨ e 2 ) \bar{e}_1,(e_1 \vee e_2) e ˉ 1 , ( e 1 ∨ e 2 ) ( e 1 ∧ e 2 ) (e_1 \wedge e_2) ( e 1 ∧ e 2 )
定義 6 - 5.2 n n n 含有 n n n 。記爲 E ( x 1 , x 2 , ⋯  , x n ) E(x_1,x_2,\cdots,x_n) E ( x 1 , x 2 , ⋯ , x n ) x 1 , x 2 , ⋯  , x n x_1,x_2,\cdots,x_n x 1 , x 2 , ⋯ , x n
定義 6 - 5.3 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ n n n E ( x 1 , x 2 , ⋯  , x n ) E(x_1,x_2,\cdots,x_n) E ( x 1 , x 2 , ⋯ , x n ) 值 是指：將 A A A x i x_i x i i = 1 , 2 , ⋯  , n i=1,2,\cdots,n i = 1 , 2 , ⋯ , n
定義 6 - 5.4 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ n n n E 1 ( x 1 , x 2 , ⋯  , x n ) E_1(x_1,x_2,\cdots,x_n) E 1 ( x 1 , x 2 , ⋯ , x n ) E 2 ( x 1 , x 2 , ⋯  , x n ) E_2(x_1,x_2,\cdots,x_n) E 2 ( x 1 , x 2 , ⋯ , x n ) n n n x i = x ~ i x_i=\widetilde{x}_i x i = x i x ~ i ∈ A \widetilde{x}_i \in A x i ∈ A E 1 ( x ~ 1 , x ~ 2 ⋯  , x ~ n ) = E 2 ( x ~ 1 , x ~ 2 ⋯  , x ~ n ) E_1(\widetilde{x}_1,\widetilde{x}_2\cdots,\widetilde{x}_n) = E_2(\widetilde{x}_1,\widetilde{x}_2\cdots,\widetilde{x}_n) E 1 ( x 1 , x 2 ⋯ , x n ) = E 2 ( x 1 , x 2 ⋯ , x n ) E 1 ( x 1 , x 2 , ⋯  , x n ) = E 2 ( x 1 , x 2 , ⋯  , x n ) E_1(x_1,x_2,\cdots,x_n) = E_2(x_1,x_2,\cdots,x_n) E 1 ( x 1 , x 2 , ⋯ , x n ) = E 2 ( x 1 , x 2 , ⋯ , x n )
定義 6 - 5.5 ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ A n A^n A n A A A ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩ n n n 布爾函數 。
定理 6 - 5.1 ⟨ { 0 , 1 } , ∨ , ∧ , ˉ ⟩ \langle \{0,1\}, \vee , \wedge,\bar{\quad} \rangle ⟨ { 0 , 1 } , ∨ , ∧ , ˉ ⟩ { 0 , 1 } n \{0,1\}^n { 0 , 1 } n { 0 , 1 } \{0,1\} { 0 , 1 }
定理 6 - 5.2 E ( x 1 , x 2 , ⋯  , x n ) E(x_1,x_2,\cdots,x_n) E ( x 1 , x 2 , ⋯ , x n ) ⟨ A , ∨ , ∧ , ˉ ⟩ \langle A,\vee,\wedge,\bar{\quad} \rangle ⟨ A , ∨ , ∧ , ˉ ⟩
