+ω法序数超运算分析

+ω法序数超运算是一种超运算型记号，作者是量子杰克，与+1法超运算类似，但区别是遇到不动点会+ω而不是+1，原因是根据分析，这样能给出更整的基本列。序数超运算可以进行许多更高级别的拓展。

定义

一个序数超运算的格式必须是$\beta \{\lambda \}\alpha$，其中β,λ,α都是正序数。

底数β只能是正整数或超限基数。若底数是超限基数，只能是${\mathrm{\Omega }}_x$，其中x是序数。定义：${\mathrm{\Omega }}_0=\omega$，对于正序数x，${\mathrm{\Omega }}_x$为第x个不可数基数（不是可数非递归序数，因为在超运算型记号中，不可数与可数非递归的效果存在本质区别）。

若底数有限，则指数α必须有限。若底数为超限基数，则指数的势必须小于等于底数的势。例如，若底数为ω，指数必须可数。若底数为${\mathrm{\Omega }}_x$，指数必须小于${\mathrm{\Omega }}_{x+1}$。

算符序数λ的势在目前版本中最多允许比底数多不超过I（首个不可达基数）。.

计算规则：

若指数为1，则值为底数。$\beta \{\lambda \}1=\beta$

若算符序数与指数均为后继序数，使用带跳不动点函数的简单迭代规则。$\beta \{\lambda +1\}(\alpha +1)=if(\beta \{\lambda \}\beta \{\lambda +1\}\alpha =\beta \{\lambda +1\}\alpha )?\beta \{\lambda \}j(\beta \{\lambda +1\}\alpha )else\beta \{\lambda \}\beta \{\lambda +1\}\alpha$

跳跃函数j(x)取$j(x)=x+\omega$，虽然可以取其他值，但根据分析，取x+ω能使基本列更整。

若指数为极限序数，值为对指数取基本列时得到的值的极限。$\beta \{\lambda \}\alpha =lim(\beta \{\lambda \}\alpha [x])$。共尾性${\mathrm{\Omega }}_x$的极限序数的基本列长度为${\mathrm{\Omega }}_x$。

若指数为后继序数，算符序数为共尾性小于等于底数的极限序数：对于$\beta ={\mathrm{\Omega }}_x,cof(\lambda )={\mathrm{\Omega }}_y,y\le x$，将指数分解为一个${\mathrm{\Omega }}_y$的倍数与一个小于${\mathrm{\Omega }}_y$的序数之和，$\alpha ={\mathrm{\Omega }}_y\times c+d$。若α<${\mathrm{\Omega }}_y$即c=0，则$\beta \{\lambda \}\alpha =\beta \{\lambda \}d=\beta \{\lambda [d]\}\beta$。若α>${\mathrm{\Omega }}_y$即c>0,$\beta \{\lambda \}\alpha =\beta \{\lambda \}({\mathrm{\Omega }}_y\times c+d)=\beta \{\lambda [d]\}j(\beta \{\lambda \}({\mathrm{\Omega }}_y\times c))$ 。

若指数为后继序数，算符序数为共尾性为底数的下一个超限基数：对于$\beta ={\mathrm{\Omega }}_x,cof(\lambda )={\mathrm{\Omega }}_{x+1}$，展开为：$\beta \{\lambda \}(\alpha +1)=if(\beta \{\lambda [\beta \{\lambda \}\alpha ]\}\beta =\beta \{\lambda \}\alpha )?\beta \{\lambda [j(\beta \{\lambda \}\alpha )]\}\beta else\beta \{\lambda [\beta \{\lambda \}\alpha ]\}\beta$。

若指数是后继序数，算符序数的共尾性大于底数的下一个超限基数，底数必须是超限基数（不能为有限数）。对于$\beta ={\mathrm{\Omega }}_x,cof(\lambda )={\mathrm{\Omega }}_{x+y+1}$，y为正序数，将指数分为ω的倍数与自然数之和：$\alpha =\omega \times c+d$。

定义：${\lambda }_1=\lambda$；${\lambda }_n$（n为大于1的正整数）= 将${\lambda }_{n-1}$的表达式中所有x>0的${\mathrm{\Omega }}_x$全部替换为${\mathrm{\Omega }}_{x+y}$。若指数有限，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}d；β{λ}1 = Ω_x{λ[Ω_(x+y)]}ω；β{λ}(n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω。若指数为超限后继序数，β{λ}α = Ω_x{λ[Ω_(x+y+1)]}(ω*c+d)；β{λ}(ω*c+1) = Ω_x{λ[Ω_(x+y)]}j(β{λ}(ω*c))；β{λ}(ω*c+n+1) = 将β{λ}n表达式中的λ_n[Ω_(x+y*n)]替换为λ_n[Ω_(x+y*n)]{λ_(n+1)[Ω_(x+y*(n+1))]}ω；若指数为极限序数，β{λ}α = lim(β{λ}(α[n]))。

分析1

分析1：SCO~HCO

序数超运算	BMS
$\omega \uparrow \uparrow \omega$	(0)(1,1) = SCO
$\omega \uparrow \uparrow (\omega +1)={\omega }^{\omega \uparrow \uparrow \omega +\omega }$	(0)(1,1)(1)(2)
$\omega \uparrow \uparrow (\omega +2)={\omega }^{{\omega }^{\omega \uparrow \uparrow \omega +\omega }}$	(0)(1,1)(1)(2,1)(2)(3)
$\omega \uparrow \uparrow (\omega +3)$	(0)(1,1)(1)(2,1)(2)(3,1)(3)(4)
$\omega \uparrow \uparrow \omega 2$	(0)(1,1)(1,1)
$\omega \uparrow \uparrow (\omega 2+1)={\omega }^{\omega \uparrow \uparrow \omega 2+\omega }$	(0)(1,1)(1,1)(1)(2)
$\omega \uparrow \uparrow \omega 3$	(0)(1,1)(1,1)(1,1)
$\omega \uparrow \uparrow {\omega }^2$	(0)(1,1)(2)
$\omega \uparrow \uparrow ({\omega }^2+\omega )$	(0)(1,1)(2)(1,1)
$\omega \uparrow \uparrow {\omega }^{2}2$	(0)(1,1)(2)(1,1)(2)
$\omega \uparrow \uparrow {\omega }^3$	(0)(1,1)(2)(2)
$\omega \uparrow \uparrow {\omega }^{\omega }$	(0)(1,1)(2)(3)
$\omega \uparrow \uparrow \uparrow 3=\omega \uparrow \uparrow \omega \uparrow \uparrow \omega$	(0)(1,1)(2)(3,1)
$\omega \uparrow \uparrow \uparrow 4$	(0)(1,1)(2)(3,1)(4)(5,1)
$\omega \uparrow \uparrow \uparrow \omega$	(0)(1,1)(2,1) = CO
$\omega \uparrow \uparrow \uparrow (\omega +1)=\omega \uparrow \uparrow (\omega \uparrow \uparrow \uparrow \omega +\omega )$	(0)(1,1)(2,1)(1,1)
$\omega \uparrow \uparrow \uparrow (\omega +2)$	(0)(1,1)(2,1)(1,1)(2)(3,1)(4,1)(3,1)
$\omega \uparrow \uparrow \uparrow \omega 2$	(0)(1,1)(2,1)(1,1)(2,1)
$\omega \uparrow \uparrow \uparrow {\omega }^2$	(0)(1,1)(2,1)(2)
$\omega \{4\}3$	(0)(1,1)(2,1)(2)(3,1)(4,1)
$\omega \{4\}\omega$	(0)(1,1)(2,1)(2,1) = LCO
$\omega \{4\}(\omega +1)=\omega \uparrow \uparrow \uparrow (\omega \{4\}\omega +\omega )$	(0)(1,1)(2,1)(2,1)(1,1)(2,1)
$\omega \{4\}\omega 2$	(0)(1,1)(2,1)(2,1)(1,1)(2,1)(2,1)
$\omega \{5\}\omega$	(0)(1,1)(2,1)(2,1)(2,1)
$\omega \{\omega \}\omega$	(0)(1,1)(2,1)(3) = HCO

分析2

分析2：HCO~FSO

序数超运算	BMS
$\omega \{\omega \}\omega$	(0)(1,1)(2,1)(3) = HCO
$\omega \{\omega \}(\omega +1)={\omega }^{\omega \{\omega \}\omega +\omega }$	(0)(1,1)(2,1)(3)(1)(2)
$\omega \{\omega \}(\omega +2)=\omega \uparrow \uparrow (\omega \{\omega \}\omega +\omega )$	(0)(1,1)(2,1)(3)(1,1)
$\omega \{\omega \}(\omega +3)=\omega \uparrow \uparrow \uparrow (\omega \{\omega \}\omega +\omega )$	(0)(1,1)(2,1)(3)(1,1)(2,1)
$\omega \{\omega \}\omega 2$	(0)(1,1)(2,1)(3)(1,1)(2,1)(3)
$\omega \{\omega \}{\omega }^2$	(0)(1,1)(2,1)(3)(2)
$\omega \{\omega +1\}3$	(0)(1,1)(2,1)(3)(2)(3,1)(4,1)(5)
$\omega \{\omega +1\}\omega$	(0)(1,1)(2,1)(3)(2,1)
$\omega \{\omega +2\}\omega$	(0)(1,1)(2,1)(3)(2,1)(2,1)
$\omega \{\omega 2\}\omega$	(0)(1,1)(2,1)(3)(2,1)(3)
$\omega \{{\omega }^2\}\omega$	(0)(1,1)(2,1)(3)(3)
$\omega \{{\omega }^{\omega }\}\omega$	(0)(1,1)(2,1)(3)(4)
$\omega \{\omega \uparrow \uparrow \omega \}\omega$	(0)(1,1)(2,1)(3)(4,1)
$\omega \{\mathrm{\Omega }\}3=\omega \{\omega \{\omega \}\omega \}\omega$	(0)(1,1)(2,1)(3)(4,1)(5,1)(6)
$\omega \{\mathrm{\Omega }\}4$	(0)(1,1)(2,1)(3)(4,1)(5,1)(6)(7,1)(8,1)(9)
$\omega \{\mathrm{\Omega }\}\omega$	(0)(1,1)(2,1)(3,1) = FSO

分析3

分析3：FSO~BHO

序数超运算	BMS
$\omega \{\mathrm{\Omega }\}\omega$	(0)(1,1)(2,1)(3,1) = FSO
$\omega \{\omega \{\mathrm{\Omega }\}\omega \}(\omega +1)=\omega \{\omega \}\omega \{\mathrm{\Omega }\}\omega$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)
$\omega \{\omega \{\mathrm{\Omega }\}\omega \}(\omega +2)=\omega \{\omega \{\omega \}\omega \}\omega \{\mathrm{\Omega }\}\omega$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6)
$\omega \{\omega \{\mathrm{\Omega }\}\omega \}\omega 2$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)
$\omega \{\omega \{\mathrm{\Omega }\}\omega +1\}\omega$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)
$\omega \{\mathrm{\Omega }\}(\omega +1)=\omega \{\omega \{\mathrm{\Omega }\}\omega +\omega \}\omega$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(2,1)(3)
$\omega \{\mathrm{\Omega }\}(\omega +2)=\omega \{\omega \{\omega \{\mathrm{\Omega }\}\omega +\omega \}\omega \}\omega$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3)(4,1)(5,1)(6,1)(4,1)(5,1)(6)(7,1)(8,1)(9,1)(5,1)(6)
$\omega \{\mathrm{\Omega }\}\omega 2$	(0)(1,1)(2,1)(3,1)(1,1)(2,1)(3,1)
$\omega \{\mathrm{\Omega }\}{\omega }^2$	(0)(1,1)(2,1)(3,1)(2)
$\omega \{\mathrm{\Omega }+1\}3$	(0)(1,1)(2,1)(3,1)(2)(3,1)(4,1)(5,1)
$\omega \{\mathrm{\Omega }+1\}\omega$	(0)(1,1)(2,1)(3,1)(2,1)
$\omega \{\mathrm{\Omega }+2\}\omega$	(0)(1,1)(2,1)(3,1)(2,1)(2,1)
$\omega \{\mathrm{\Omega }+\omega \}\omega$	(0)(1,1)(2,1)(3,1)(2,1)(3)
$\omega \{\mathrm{\Omega }2\}3=\omega \{\mathrm{\Omega }+\omega \{\mathrm{\Omega }+\omega \}\omega \}\omega$	(0)(1,1)(2,1)(3,1)(2,1)(3)(4,1)(5,1)(4,1)(5)
$\omega \{\mathrm{\Omega }2\}\omega$	(0)(1,1)(2,1)(3,1)(2,1)(3,1)
$\omega \{\mathrm{\Omega }\omega \}\omega$	(0)(1,1)(2,1)(3,1)(3)
$\omega \{{\mathrm{\Omega }}^2\}3=\omega \{\mathrm{\Omega }\times \omega \{\mathrm{\Omega }\omega \}\omega \}\omega$	(0)(1,1)(2,1)(3,1)(3)(4,1)(5,1)(5)
$\omega \{{\mathrm{\Omega }}^2\}\omega$	(0)(1,1)(2,1)(3,1)(3,1) = ACO
$\omega \{{\mathrm{\Omega }}^2+1\}\omega$	(0)(1,1)(2,1)(3,1)(3,1)(2,1)
$\omega \{{\mathrm{\Omega }}^{2}2\}\omega$	(0)(1,1)(2,1)(3,1)(3,1)(2,1)(3,1)(3,1)
$\omega \{{\mathrm{\Omega }}^2\omega \}\omega$	(0)(1,1)(2,1)(3,1)(3,1)(3)
$\omega \{{\mathrm{\Omega }}^3\}\omega$	(0)(1,1)(2,1)(3,1)(3,1)(3,1)
$\omega \{{\mathrm{\Omega }}^{\omega }\}\omega$	(0)(1,1)(2,1)(3,1)(4) = SVO
$\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }}\}3=\omega \{{\mathrm{\Omega }}^{\omega \{{\mathrm{\Omega }}^{\omega }\}\omega }\}\omega$	(0)(1,1)(2,1)(3,1)(4)(5,1)(6,1)(7,1)(8)
$\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1) = LVO
$\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }}+1\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(2,1)
$\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }+1}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(3,1)
$\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }2}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(3,1)(4,1)
$\omega \{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^2}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(4,1)
$\omega \{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\omega }}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(5)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}3=\omega \{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(5,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}4=\omega \{\mathrm{\Omega }\uparrow \uparrow 4\}\omega$	(0)(1,1)(2,1)(3,1)(4,1)(5,1)(6,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1)(2,2) = BHO

分析4

分析4：BHO~(0)(1,1)(2,2)(3,2)(4)

序数超运算	BMS
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1)(2,2) = BHO
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}(\omega +1)=\omega \{\mathrm{\Omega }\}(\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega +\omega )$	(0)(1,1)(2,2)(1,1)(2,1)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}(\omega +1)=\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }}\}(\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega +\omega )$	(0)(1,1)(2,2)(1,1)(2,1)(3,1)(4,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega 2$	(0)(1,1)(2,2)(1,1)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}{\omega }^2$	(0)(1,1)(2,2)(2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega +1\}3$	(0)(1,1)(2,2)(2)(3,1)(4,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega +1\}\omega$	(0)(1,1)(2,2)(2,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega +\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(2,1)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \times 2\}\omega$	(0)(1,1)(2,2)(2,1)(3,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \times \omega \}\omega$	(0)(1,1)(2,2)(2,1)(3,2)(3)
$\omega \{\mathrm{\Omega }\uparrow \uparrow (\omega +1)\}\omega =\omega \{{\omega }^{\mathrm{\Omega }\uparrow \uparrow \omega +\omega }\}\omega$	(0)(1,1)(2,2)(2,1)(3,2)(3)(4)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega 2\}\omega$	(0)(1,1)(2,2)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega 3\}\omega$	(0)(1,1)(2,2)(2,2)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow {\omega }^2\}\omega$	(0)(1,1)(2,2)(3)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}3=\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}\omega \}\omega$	(0)(1,1)(2,2)(3)(4,1)(5,2)(6)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow (\mathrm{\Omega }+\omega )\}\omega$	(0)(1,1)(2,2)(3,1)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }2\}\omega$	(0)(1,1)(2,2)(3,1)(2,2)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow {\mathrm{\Omega }}^2\}\omega$	(0)(1,1)(2,2)(3,1)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1)(2,2)(3,1)(4,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow 3\}\omega$	(0)(1,1)(2,2)(3,1)(4,2)(5,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow \omega \}\omega$	(0)(1,1)(2,2)(3,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow (\omega +1)\}\omega =\omega \{\mathrm{\Omega }\uparrow \uparrow (\mathrm{\Omega }\uparrow \uparrow \uparrow \omega +\omega )\}\omega$	(0)(1,1)(2,2)(3,2)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow \omega 2\}\omega$	(0)(1,1)(2,2)(3,2)(2,2)(3,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow {\omega }^2\}\omega$	(0)(1,1)(2,2)(3,2)(3)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow \mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(3,1)
$\omega \{\mathrm{\Omega }\{4\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(3,2)
$\omega \{\mathrm{\Omega }\{\omega \}4\}\omega =\omega \{\mathrm{\Omega }\{4\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(3,2)(3,1)
$\omega \{\mathrm{\Omega }\{5\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(3,2)(3,2)
$\omega \{\mathrm{\Omega }\{\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)

分析5

分析：(0)(1,1)(2,2)(3,2)(4)~(0)(1,1)(2,2)(3,2)(4,2)

序数超运算	BMS
$\omega \{\mathrm{\Omega }\{\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)
$\omega \{\mathrm{\Omega }\{\omega \}(\omega +1)\}\omega =\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }\{\omega \}\omega +\omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4)(2,1)(3,2)(3)(4)
$\omega \{\mathrm{\Omega }\{\omega \}(\omega +2)\}\omega =\omega \{\mathrm{\Omega }\uparrow \uparrow (\mathrm{\Omega }\{\omega \}\omega +\omega )\}\omega$	(0)(1,1)(2,2)(3,2)(4)(2,2)
$\omega \{\mathrm{\Omega }\{\omega \}\omega 2\}\omega$	(0)(1,1)(2,2)(3,2)(4)(2,2)(3,2)(4)
$\omega \{\mathrm{\Omega }\{\omega \}{\omega }^2\}\omega$	(0)(1,1)(2,2)(3,2)(4)(3)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}\omega \}\omega =\omega \{\mathrm{\Omega }\{\omega \}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4)(3,1)
$\omega \{\mathrm{\Omega }\{\omega +1\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(3,2)
$\omega \{\mathrm{\Omega }\{\omega 2\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(3,2)(4)
$\omega \{\mathrm{\Omega }\{{\omega }^2\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(4)
$\omega \{\mathrm{\Omega }\{\omega \uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(5,1)
$\omega \{\mathrm{\Omega }\{\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(5,1)(6,2)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}(\mathrm{\Omega }+1)\}\omega =\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }+\omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(2,1)(3,2)(3)(4)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}(\mathrm{\Omega }+2)\}\omega =\omega \{\mathrm{\Omega }\uparrow \uparrow (\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }+\omega )\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(2,2)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}(\mathrm{\Omega }+\omega )\}\omega =\omega \{\mathrm{\Omega }\{\omega \}(\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }+\omega )\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }2\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(2,2)(3,2)(4,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}{\mathrm{\Omega }}^2\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }+1\}3\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3,1)(4,2)(5,3)(6,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }+1\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3,2)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }+\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }2\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(3,2)(4,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(4)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}^2\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(4,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,1)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\uparrow \uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\{\omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}3=\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}3\}\omega =\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}4=\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}4\}\omega$	(0)(1,1)(2,2)(3,2)(4,1)(5,2)(6,2)(7,1)(8,2)(9,2)(10,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4,2)

预期强度

预期强度（已进行部分分析，待填写）

序数超运算	BMS
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{\omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,4)
$\omega \{{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1) = BO
$\omega \{{\mathrm{\Omega }}_{2}2\}\omega$	(0)(1,1,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+1)\}\omega$	(0)(1,1,1)(2,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }\uparrow \uparrow \omega )\}\omega$	(0)(1,1,1)(2,1)(3,2) = TFBO
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega )\}\omega$	(0)(1,1,1)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_2^2\}\omega$	(0)(1,1,1)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1) = BIO
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\omega \}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2) = EBO
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2) = JO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1) = SIO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = SMO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = SKO
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2) = SSO
$\omega \{{\mathrm{\Omega }}_2\{\mathrm{\Omega }\}\mathrm{\Omega }\times \omega \}\omega$	(0)(1,1,1)(2,2)(3,2)(4,1)(2) = LSO
$\omega \{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_2\}{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,2)(3,2)(4,1,1) = APO
$\omega \{{\mathrm{\Omega }}_3\}\omega$	(0)(1,1,1)(2,2,1) = BGO
$\omega \{{\mathrm{\Omega }}_4\}\omega$	(0)(1,1,1)(2,2,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_{\omega }\}\omega$	(0)(1,1,1)(2,2,1)(3) = SDO
$\omega \{{\mathrm{\Omega }}_{\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_{{\mathrm{\Omega }}_2}\}\omega$	(0)(1,1,1)(2,2,1)(3,1,1)
$\omega \{\mathrm{\Phi }(1,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2) = LDO
$psd.\omega \{I\}\omega =\omega \{{\psi }_I(X)\}\omega ct.\psi (X)$	(0)(1,1,1)(2,2,1)(3,2,1)
$psd.\omega \{\psi (X)\}\omega ct.\psi (X)$	(0)(1,1,1)(2,2,2) = pfec LRO