+ω法序数超运算分析

以下是+ω法序数超运算的分析。

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)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \times \mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(2,1)(3,2)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow (\omega +1)\}\omega =\omega \{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +\omega }\}\omega$	(0)(1,1)(2,2)(2,1)(3,2)(3,1)(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5：(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)
$\omega \{\mathrm{\Omega }\{\omega \{\mathrm{\Omega }\{\omega \}\omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,2)(4)(5,1)(6,2)(7)
$\omega \{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(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)

分析6

分析6：(0)(1,1)(2,2)(3,2)(4,2)~(0)(1,1)(2,2)(3,3)

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

分析7

分析7：(0)(1,1)(2,2)(3,3)~BO

序数超运算	BMS
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega +1\}\omega$	(0)(1,1)(2,2)(3,3)(2,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \times 2\}\omega$	(0)(1,1)(2,2)(3,3)(2,1)(3,2)(4,3)
$\omega \{\mathrm{\Omega }\uparrow \uparrow (\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega )\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(2,2)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \uparrow (\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega )\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(2,2)(3,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}(\omega +1)\}\omega =\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}(\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega )\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(2,2)(3,2)(4,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega 2\}\omega$	(0)(1,1)(2,2)(3,3)(2,2)(3,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}{\omega }^2\}\omega$	(0)(1,1)(2,2)(3,3)(3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,3)(3,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +1\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times 2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times \omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,3)(5)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times \mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,3)(5,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +1}\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,3)(5,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow (\omega +1)\}\omega \}\omega =\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega }\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,2)(4,3)(5,2)(6)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega 2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(3,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow {\omega }^2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,3)(4,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow {\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow 3\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,2)(5,3)(6,2)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow \omega \}\omega \}\omega$	(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)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{\mathrm{\Omega }\}\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,1)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_2\}{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,2)
$\omega \{{\mathrm{\Omega }}_2\}4=\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_{3}2\}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,3)(4,3)(5,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3^2\}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,3)(5,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3^{{\mathrm{\Omega }}_3}\}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,3)(5,3)(6,3)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,4)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\uparrow \uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,4)(5,4)
$\omega \{{\mathrm{\Omega }}_2\}5=\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_4\}\omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,4)(5,4)(6,4)
$\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_4\uparrow \uparrow \omega \}\omega \}\omega \}\omega \}\omega$	(0)(1,1)(2,2)(3,3)(4,4)(5,5)
$\omega \{{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1) = BO

分析8

分析8：BO~(0)(1,1,1)(2,1)

序数超运算	BMS
$\omega \{{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1) = BO
$\omega \uparrow \uparrow (\omega \{{\mathrm{\Omega }}_2\}\omega +\omega )$	(0)(1,1,1)(1,1)
$\omega \{{\mathrm{\Omega }}_2\}(\omega +1)=\omega \{\mathrm{\Omega }\}(\omega \{{\mathrm{\Omega }}_2\}\omega +\omega )$	(0)(1,1,1)(1,1)(2,1)(3,1)
$\omega \{\mathrm{\Omega }\uparrow \uparrow \omega \}(\omega \{{\mathrm{\Omega }}_2\}\omega +\omega )$	(0)(1,1,1)(1,1)(2,2)
$\omega \{{\mathrm{\Omega }}_2\}(\omega +2)=\omega \{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}(\omega \{{\mathrm{\Omega }}_2\}\omega +\omega )$	(0)(1,1,1)(1,1)(2,2)(3,2)(4,2)
$\omega \{{\mathrm{\Omega }}_2\}\omega 2$	(0)(1,1,1)(1,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_2\}{\omega }^2$	(0)(1,1,1)(1,1)(2,2,1)(2)
$\omega \{{\mathrm{\Omega }}_2+1\}3$	(0)(1,1,1)(1,1)(2,2,1)(2)(3,1,1)
$\omega \{{\mathrm{\Omega }}_2+1\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2+2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2+\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3)
$\omega \{{\mathrm{\Omega }}_2+\omega \{{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3)(4,1,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }+1\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2+{\mathrm{\Omega }}^2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\uparrow \uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,2)(5)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,2)(5,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,2)(5,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2)(4,3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega +1}\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}(\omega +1)\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,1)(5,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}(\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega +\omega )\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega 2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}{\omega }^2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2,1)(5)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2,1)(5,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2,1)(5,1)(6,2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}3\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,1)(3,2,1)(3,1)(4,2,1)(5,1)(6,2,1)(7,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega +1\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega +\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega \times 2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)(3,2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2+{\mathrm{\Omega }}^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega +1}\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)(3,2,1)(3,2)(3,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}(\omega +1)\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)(3,2,1)(3,2)(3,1)(4,2,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+1\}\omega 2\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,1)(3,2,1)(3,2)(3,1)(4,2,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+2\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(2,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,1)
$\omega \{{\mathrm{\Omega }}_{2}2\}3=\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,2)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3)(4,3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3)(4,3)(5,3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3)(4,4)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4+1\}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)
$\omega \{{\mathrm{\Omega }}_{2}2\}4=\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4+{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)(4,3)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4+{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)(4,4)
$\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4+{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_5\}\omega \}\omega \}\omega \}\omega$	(0)(1,1,1)(1,1)(2,2,1)(2,2)(3,3,1)(3,3)(4,4,1)
$\omega \{{\mathrm{\Omega }}_{2}2\}\omega$	(0)(1,1,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\}(\omega \{{\mathrm{\Omega }}_{2}2\}\omega +\omega )$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_{2}2\}(\omega +1)=\omega \{{\mathrm{\Omega }}_2+\mathrm{\Omega }\}(\omega \{{\mathrm{\Omega }}_{2}2\}\omega +\omega )$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_{2}2\}\omega 2$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_{2}2+1\}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,1)
$\omega \{{\mathrm{\Omega }}_{2}2+\mathrm{\Omega }\}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_{2}2+\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_{2}2+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_{2}2+\mathrm{\Omega }\{{\mathrm{\Omega }}_{3}2\}\omega \}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,1)(3,2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_{2}2+\mathrm{\Omega }\{{\mathrm{\Omega }}_{3}2+1\}\omega \}\omega$	(0)(1,1,1)(1,1,1)(1,1)(2,2,1)(2,2,1)(2,2)
$\omega \{{\mathrm{\Omega }}_{2}3\}\omega$	(0)(1,1,1)(1,1,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\omega \}\omega$	(0)(1,1,1)(2)
$\omega \{{\mathrm{\Omega }}_2\times \omega \{{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(2)(3,1,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }\}3$	(0)(1,1,1)(2)(3,1,1)(4)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)

分析9

分析9：(0)(1,1,1)(2,1)~TFBO

序数超运算	BMS
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }\}\omega 2$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+1\}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,1)(3,2,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }+1\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }+{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }+{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\mathrm{\Omega }+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\mathrm{\Omega }\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2)(3,3,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times (\mathrm{\Omega }+1)\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times (\mathrm{\Omega }+2)\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }2\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(2,2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times {\mathrm{\Omega }}^2\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times {\mathrm{\Omega }}^{\mathrm{\Omega }}\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }\uparrow \uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2)(5,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2)(5,2)(6,2)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2)(5,3)
$\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,1)(4,2,1)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+1)\}3=\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+1)\}4=\omega \{{\mathrm{\Omega }}_2\mathrm{\Omega }+\mathrm{\Omega }\{{\mathrm{\Omega }}_3{\mathrm{\Omega }}_2+{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4{\mathrm{\Omega }}_3\}\}\omega \}\omega$	(0)(1,1,1)(2,1)(1,1)(2,2,1)(3,2)(2,2)(3,3,1)(4,3)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+1)\}\omega$	(0)(1,1,1)(2,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+2)\}\omega$	(0)(1,1,1)(2,1)(1,1,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\times (\mathrm{\Omega }+\omega )\}\omega$	(0)(1,1,1)(2,1)(1,1,1)(2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }2\}\omega$	(0)(1,1,1)(2,1)(1,1,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\omega \}\omega$	(0)(1,1,1)(2,1)(2)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^2\}\omega$	(0)(1,1,1)(2,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2\times ({\mathrm{\Omega }}^2+1)\}\omega$	(0)(1,1,1)(2,1)(2,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\omega }\}\omega$	(0)(1,1,1)(2,1)(3)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2\times ({\mathrm{\Omega }}^{\mathrm{\Omega }}+1)\}\omega$	(0)(1,1,1)(2,1)(3,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1)(3,2) = TFBO

分析10

分析10：TFBO~EBO

序数超运算	BMS
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1)(3,2) = TFBO
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega \times 2\}\omega$	(0)(1,1,1)(2,1)(3,2)(1,1,1)(2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega \times \omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(2)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +1}\}\omega$	(0)(1,1,1)(2,1)(3,2)(2,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow (\omega +1)\}\omega =\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +\omega }\}\omega$	(0)(1,1,1)(2,1)(3,2)(2,1)(3)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega \times 2}\}\omega$	(0)(1,1,1)(2,1)(3,2)(2,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega \times \omega }\}\omega$	(0)(1,1,1)(2,1)(3,2)(3)
$\omega \{{\mathrm{\Omega }}_2\times {\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +1}}\}\omega$	(0)(1,1,1)(2,1)(3,2)(3,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow (\omega +2)\}\omega$	(0)(1,1,1)(2,1)(3,2)(3,1)(4)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega 2\}\omega$	(0)(1,1,1)(2,1)(3,2)(3,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow {\omega }^2\}\omega$	(0)(1,1,1)(2,1)(3,2)(4)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)(3,2)(4,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(4,1)(5,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \uparrow 3\}\omega$	(0)(1,1,1)(2,1)(3,2)(4,1)(5,2)(6,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(4,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(4,2)(5)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1)(3,2)(4,2)(5,1)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(4,2)(5,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2)(4,3)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_2^2\}3=\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3{\mathrm{\Omega }}_2\}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\times {\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3)
$\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\times {\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)
$\omega \{{\mathrm{\Omega }}_2^2\}4=\omega \{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\times {\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4{\mathrm{\Omega }}_3\}\omega \}\omega \}\omega$	(0)(1,1,1)(2,1)(3,2,1)(4,2)(5,3,1)(6,3)
$\omega \{{\mathrm{\Omega }}_2^2\}\omega$	(0)(1,1,1)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^3\}\omega$	(0)(1,1,1)(2,1,1)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3)
$\omega \{{\mathrm{\Omega }}_2^{\omega \{{\mathrm{\Omega }}_2\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3)(4,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\}3$	(0)(1,1,1)(2,1,1)(3)(4,1,1)(5,1,1)(6)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1) = BIO
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}+{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}2\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(1,1,1)(2,1,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\omega \}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2) = EBO

分析11

分析11：EBO~SSO

序数超运算	BMS
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\omega \}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2) = EBO
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }+1}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }+\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(2,1,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2) = JO
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega +1}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega \times 2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(2,1,1)(3,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega \times \omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(3)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +1}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +1}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow (\omega +1)}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(3,1)(4)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega \times 2}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(3,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega \times \omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(4)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}^{{\mathrm{\Omega }}^{\mathrm{\Omega }\uparrow \uparrow \omega +1}}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(4,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow (\omega +2)}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(4,1)(5)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega 2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(4,2)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow {\omega }^2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \uparrow \omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,2)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{\omega \}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,2)(6)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{\mathrm{\Omega }\}\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,2)(6,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,2)(6,2)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2)(5,3)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3^2\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)
$\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3^{\mathrm{\Omega }}\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\}3=\omega \{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3^{{\mathrm{\Omega }}_2}\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1)(4,2,1)(5,2,1)(6,2)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1) = SIO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2+1}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_{2}2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(2,1,1)(3,1,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3) = MBO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\mathrm{\Omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\mathrm{\Omega }}\omega \}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(2) = TBO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\uparrow \uparrow \omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2) = SRO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\times \mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega }\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1)(4,2,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^2}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1) = SMO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^3}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(3,1,1)(3,1,1) = SNO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\mathrm{\Omega }}}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\uparrow \uparrow \omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2) = RO
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2^{\mathrm{\Omega }\{{\mathrm{\Omega }}_3\}\omega }}\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1)(5,2,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow 3\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1) = SKO
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow 3|\mathrm{\Omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1)(6,2) = DO
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow 4\}\omega$	(0)(1,1,1)(2,1,1)(3,1,1)(4,1,1)(5,1,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2) = SSO

预期强度

预期强度（已进行部分分析，待填写）

序数超运算	BMS
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2) = SSO
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,2)(1,1,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times 2\}\omega$	(0)(1,1,1)(2,2)(1,1,1)(2,2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times \omega \}\omega$	(0)(1,1,1)(2,2)(2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times \mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,2)(2,1)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +1}\}\omega$	(0)(1,1,1)(2,2)(2,1,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow (\omega +1)\}\omega =\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\uparrow \uparrow \omega +\omega }\}\omega$	(0)(1,1,1)(2,2)(2,1,1)(3)
$\omega \{{\mathrm{\Omega }}_2^{{\mathrm{\Omega }}_2\uparrow \uparrow \omega \times 2}\}\omega$	(0)(1,1,1)(2,2)(2,1,1)(3,2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \omega 2\}\omega$	(0)(1,1,1)(2,2)(2,2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow {\omega }^2\}\omega$	(0)(1,1,1)(2,2)(3)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,2)(3,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow {\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,2)(3,1,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow {\mathrm{\Omega }}_2\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2)(3,1,1)(4,2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow 3\}\omega$	(0)(1,1,1)(2,2)(3,1,1)(4,2)(5,1,1)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2)(3,2)
$\omega \{{\mathrm{\Omega }}_2\uparrow \uparrow \uparrow {\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,2)(3,2)(3,1,1)
$\omega \{{\mathrm{\Omega }}_2\{4\}\omega \}\omega$	(0)(1,1,1)(2,2)(3,2)(3,2)
$\omega \{{\mathrm{\Omega }}_2\{\omega \}\omega \}\omega$	(0)(1,1,1)(2,2)(3,2)(4)
$\omega \{{\mathrm{\Omega }}_2\{\mathrm{\Omega }\}\mathrm{\Omega }\}\omega$	(0)(1,1,1)(2,2)(3,2)(4,1)
$\omega \{{\mathrm{\Omega }}_2\{\mathrm{\Omega }\}\mathrm{\Omega }\times \omega \}\omega$	(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
$\omega \{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\}\omega \}\omega$	(0)(1,1,1)(2,2)(3,2)(4,2)
$\omega \{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,2)(3,3)
$\omega \{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\}\omega \}\omega$	(0)(1,1,1)(2,2)(3,3,1)
$\omega \{{\mathrm{\Omega }}_2\{{\mathrm{\Omega }}_4\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,2)(3,3,1)(4,4)
$\omega \{{\mathrm{\Omega }}_3\}\omega$	(0)(1,1,1)(2,2,1) = BGO
$\omega \{{\mathrm{\Omega }}_3+{\mathrm{\Omega }}_2\}\omega$	(0)(1,1,1)(2,2,1)(1,1,1)
$\omega \{{\mathrm{\Omega }}_{3}2\}\omega$	(0)(1,1,1)(2,2,1)(1,1,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_3\omega \}\omega$	(0)(1,1,1)(2,2,1)(2)
$\omega \{{\mathrm{\Omega }}_3^2\}\omega$	(0)(1,1,1)(2,2,1)(2,1,1)(3,2,1)
$\omega \{{\mathrm{\Omega }}_3\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2,1)(2,2)
$\omega \{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_4\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(2,2)(3,2)(4,2)
$\omega \{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_4\uparrow \uparrow \omega \}\omega \}\omega$	(0)(1,1,1)(2,2,1)(2,2)(3,3)
$\omega \{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_5\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(2,2)(3,3,1)
$\omega \{{\mathrm{\Omega }}_3\{{\mathrm{\Omega }}_6\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(2,2)(3,3,1)(4,4,1)
$\omega \{{\mathrm{\Omega }}_4\}\omega$	(0)(1,1,1)(2,2,1)(2,2,1)
$\omega \{{\mathrm{\Omega }}_{\omega }\}\omega$	(0)(1,1,1)(2,2,1)(3) = SDO
$\omega \{{\mathrm{\Omega }}_{\omega }\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2,1)(3)(2,2)
$\omega \{{\mathrm{\Omega }}_{\omega +1}\}\omega$	(0)(1,1,1)(2,2,1)(3)(2,2,1)
$\omega \{{\mathrm{\Omega }}_{\mathrm{\Omega }}\}\omega$	(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)
$\omega \{{\mathrm{\Omega }}_{{\mathrm{\Omega }}_{\omega }}\}\omega$	(0)(1,1,1)(2,2,1)(3,1,1)(4,2,1)(5)
$\omega \{\mathrm{\Phi }(1,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2) = LDO

预期强度（超下标运算）

预期强度（超下标运算）

序数超运算	BMS
$\omega \{\omega \downarrow \downarrow \omega \}\omega =\omega \{\mathrm{\Phi }(1,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2) = LDO
$\omega \{(\omega \downarrow \downarrow \omega )\uparrow \uparrow \omega \}\omega =\omega \{\mathrm{\Phi }(1,0)\uparrow \uparrow \omega \}\omega$	(0)(1,1,1)(2,2,1)(3,2)(2,2)
$\omega \{{\mathrm{\Omega }}_{\omega \downarrow \downarrow \omega +1}\}\omega =\omega \{{\mathrm{\Omega }}_{\mathrm{\Phi }(1,0)+1}\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(2,2,1)
$\omega \{\omega \downarrow \downarrow (\omega +1)\}\omega =\omega \{{\mathrm{\Omega }}_{\mathrm{\Phi }(1,0)+\omega }\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(2,2,1)(3)
$\omega \{\omega \downarrow \downarrow \omega 2\}\omega =\omega \{\mathrm{\Phi }(1,1)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(2,2,1)(3,2)
$\omega \{\omega \downarrow \downarrow {\omega }^2\}\omega =\omega \{\mathrm{\Phi }(1,\omega )\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(3)
$\omega \{\omega \downarrow \downarrow \mathrm{\Omega }\}\omega =\omega \{\mathrm{\Phi }(1,\mathrm{\Omega })\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(3,1)
$\omega \{\omega \downarrow \downarrow {\mathrm{\Omega }}_2\}\omega =\omega \{\mathrm{\Phi }(1,{\mathrm{\Omega }}_2)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(3,1,1)
$\omega \{\omega \downarrow \downarrow \downarrow 3\}\omega =\omega \{\mathrm{\Phi }(1,\mathrm{\Phi }(1,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(3,1,1)(4,2,1)(5,2)
$\omega \{\omega \downarrow \downarrow \downarrow \omega \}\omega =\omega \{\mathrm{\Phi }(2,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(3,2)
$\omega \{\omega \{\{\omega \}\}\omega \}\omega =\omega \{\mathrm{\Phi }(\omega ,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4)
$\omega \{\omega \{\{\mathrm{\Omega }\}\}\omega \}\omega =\omega \{\mathrm{\Phi }(\mathrm{\Omega },0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,1)
$\omega \{\omega \{\{{\mathrm{\Omega }}_2\}\}\omega \}\omega =\omega \{\mathrm{\Phi }({\mathrm{\Omega }}_2,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,1,1)
$\omega \{\omega \{\{I\}\}3\}\omega =\omega \{\mathrm{\Phi }(\mathrm{\Phi }(\omega ,0),0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,1,1)(5,2,1)(6,2)(7)
$\omega \{\omega \{\{I\}\}\omega \}\omega =\omega \{\mathrm{\Phi }(1,0,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)
$\omega \{\omega \{\{I+1\}\}\omega \}\omega =\omega \{\mathrm{\Phi }(1,1,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)(3,2)
$\omega \{\omega \{\{I2\}\}\omega \}\omega =\omega \{\mathrm{\Phi }(2,0,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)(3,2)(4,2)
$\omega \{\omega \{\{I^2\}\}\omega \}\omega =\omega \{\mathrm{\Phi }(1,0,0,0)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)(4,2)
$\omega \{\omega \{\{I^{\omega }\}\}\omega \}\omega =\omega \{{\psi }_I(I^{I^{\omega }})\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)(5)
$\omega \{\omega \{\{I^I\}\}\omega \}\omega =\omega \{{\psi }_I(I^{I^I})\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,2)(5,2)
$\omega \{\omega \{\{I\uparrow \uparrow \omega \}\}\omega \}\omega =\omega \{{\psi }_I({\mathrm{\Omega }}_{I+1})\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3)
$\omega \{\omega \{\{I\uparrow \uparrow \uparrow \omega \}\}\omega \}\omega =\omega \{{\psi }_I({\mathrm{\Omega }}_{I+1}^2)\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3)(5,3)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I+1}\}\omega \}\}\omega \}\omega =\omega \{{\psi }_I({\mathrm{\Omega }}_{I+1}^{{\mathrm{\Omega }}_{I+1}})\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3)(5,3)(6,3)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I+1}\uparrow \uparrow \omega \}\omega \}\}\omega \}\omega =\omega \{{\psi }_I({\mathrm{\Omega }}_{I+2})\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3)(5,4)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I+2}\}\omega \}\}\omega \}\omega =\omega \{{\psi }_I({\mathrm{\Omega }}_{I+\omega })\}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3,1)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I+3}\}\omega \}\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3,1)(5,4,1)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I+\omega }\}\omega \}\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3,1)(5,4,1)(6)
$\omega \{\omega \{\{I\{{\mathrm{\Omega }}_{I2}\}\omega \}\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3,1)(5,4,1)(6,2)
$\omega \{\omega \{\{I\{\omega \downarrow \downarrow (I+\omega )\}\omega \}\}\omega \}\omega$	(0)(1,1,1)(2,2,1)(3,2)(4,3,1)(5,4,1)(6,4)
$\omega \{I\}\omega =\omega \{\omega \{\{X\to I\{\omega \{\{X\}\}(I+\omega )\}\omega fp\}\}\omega \}\omega =\omega \{{\psi }_I(X)\}\omega ct.\psi (X)$	(0)(1,1,1)(2,2,1)(3,2,1)
$psd.\omega \{X\}\omega ct.\psi (X)$	(0)(1,1,1)(2,2,2) = pfec LRO