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❖r❞❡r❡❞ ❋❛❝❡ ❙tr✉❝t✉r❡s ❛♥❞ ▼❛♥②✲t♦✲♦♥❡ ❈♦♠♣✉t❛❞s ▼❛r❡❦ ❩❛✇❛❞♦✇s❦✐ ❏✉♥❡ ✷✷✱ ✷✵✵✼✱ ❈❛r✈♦❡✐r♦✱ P♦rt✉❣❛❧

▼❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s ❛r❡ ω ✲❝❛t❡❣♦r✐❡s t❤❛t ❛r❡ ❢r❡❡ ✬❧❡✈❡❧✇✐s❡✬ ✐✳❡✳ ✇❡ ❛❞❥♦✐♥ n + 1 ❞✐♠❡♥✲ s✐♦♥❛❧ ❣❡♥❡r❛t♦rs ✭❂ ✐♥❞❡ts ✮ ❛♥❞ ✇❡ ✜① t❤❡✐r ❞♦♠❛✐♥s ❛♥❞ ❝♦❞♦♠❛✐♥s ♦♥❧② ❛❢t❡r ❣❡♥❡r❛t✐♦♥ ♦♥ ❛❧❧ n ❞✐♠❡♥s✐♦♥❛❧ ❝❡❧❧✳ ▼♦r❡♦✈❡r ✇❡ ✐♥✲ s✐sts t❤❛t t❤❡ ❝♦❞♦♠❛✐♥ ♦❢ t❤❡ ✐♥❞❡ts ❛r❡ ✐♥❞❡ts ❛❣❛✐♥ ✭❂ ♠❛♥②✲t♦✲♦♥❡✮✳ ❲❤② ♠❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s❄ ❚❤❡② s❡❡♠ t♦ ❜❡ ✐♥ t❤❡ ❝❡♥t❡r ♦❢ t✇♦ ❛♣♣r♦❛❝❤❡s t♦ ✇❡❛❦ ω ✲❝❛t❡❣♦r✐❡s✿ ♠✉❧t✐t♦♣✐❝ ❛♥❞ ♦♣❡t♦♣✐❝✳ ❚❤❡ ❝❛t❡❣♦r② Comp m/ 1 ♦❢ ♠❛♥②✲t♦✲♦♥❡ ❝♦♠✲ ♣✉t❛❞s ❛♥❞ ❝♦♠♣✉t❛❞s ♠❛♣ ✐s ❡q✉✐✈❛❧❡♥t t♦ t❤❡ ❝❛t❡❣♦r② ♦❢ ♠✉❧t✐t♦♣✐❝ s❡ts MltSets ✳✳✳ ✳✳✳❛♥❞ ♣r♦❜❛❜❧② t♦ t❤❡ ❝❛t❡❣♦r② ♦❢ ♦♣❡t♦♣✐❝ s❡ts✱ ❛s ✇❡❧❧✳ ✶

❖r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ❛r❡ ❝♦♠❜✐♥❛t♦r✐❛❧ str✉❝t✉r❡s ❞❡s❝r✐❜✐♥❣ t❤❡ ✬t②♣❡s✬ ♦❢ ✭❛❧❧✮ ❝❡❧❧s ✐♥ ♠❛♥②✲t♦✲♦♥❡ ❝♦♠♣✉t❛❞s✳ ❊①❛♠♣❧❡s✳ • � ✒ ❅ x 6 � ❅ ⇓ � ❅ � ❅ ❘ • • ✲ ✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏✏ ✶ � ✒ ❅ ⇓ x 2 � ❅ x 3 ⇓ a 1 � ❅ � ❅ ❘ x 0 s 0 • • ✲ ✲ x 1 ❉♦♠❛✐♥s✿ δ ( a 1 ) = { x 2 , x 3 } ❈♦❞♦♠❛✐♥s✿ γ ( a 1 ) = x 1 ❚✇♦ ♦r❞❡rs✳ ▲♦✇❡r✿ x 1 < − x 0 ❛s γ ( x 1 ) ∈ δ ( x 0 ) ✱ ✳✳✳ ❛♥❞ ❜② tr❛♥s✐t✐✈✐t②✿ x 6 < − x 0 ❯♣♣❡r✿ x 3 < + x 1 ❛s x 3 ∈ δ ( a 3 ) ❛♥❞ γ ( a 3 ) = x 1 ✱ ✳✳✳ ❛♥❞ ❜② tr❛♥s✐t✐✈✐t②✿ x 6 < − x 1 ✷

❍♦✇❡✈❡r ✇❡ ♠❛② ❤❛✈❡ ❡♠♣t②✲❞♦♠❛✐♥ ❢❛❝❡s ✳✳✳ ❛♥❞ ❤❡♥❝❡ ❡♠♣t② ❢❛❝❡s ❛♥❞ ❧♦♦♣s s 2 ✒ � ❅ x 7 � x 6 � ❅ ⇓ a 5 ❘ ❅ s 3 s 1 ✲ x 5 ✁ ✻ � ✒ ❆ ❑ ❅ ✁ ❆ � ❅ ✁ ❆ � ❅ ⇓ ⇓ ✁ ❆ � ❅ x 8 ✁ ❆ x 2 ✖✕ a 4 ✖✕ a 3 x 3 � ❅ ⇓ a 1 � ❅ x 4 ✔ ✗ � ❅ � ✟ ❍❍❍ ❅ ✟ a 6 x 9 a 2 � ✟ ✕ ✖ ❅ ❘ ✙ ✟ ❍ ❥ ⇒ ⇐ x 1 s 4 s 0 ✲ x 0 δ ( a 2 ) = 1 s 0 ✱ γ ( a 2 ) = x 1 ❛♥❞ ♠♦r❡ ❧♦♦♣s s s ✁ ❆ ❑ ✁ ❆ ❑ ✁ ✁ ✻ ❆ ✁ ❆ ❑ ❆ ✁ ❆ β ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ⇓ ⇒ ⇓ = ✁ ❆ ✁ ❆ ✁ ❆ ⇓ b 0 ✁ ❆ ✖✕ b 3 ✖✕ b 2 ✁ ❆ ✁ ❆ y 2 y 1 ⇓ b 1 y 0 y 0 ✫ ✪ ✫ ✪ ❲❤❛t ❛❜♦✉t ❧♦✇❡r ♦r❞❡r ♦❢ x 3 ❛♥❞ x 4 ❄✳✳✳ ♦r y 2 ❛♥❞ y 3 ❄ ❲❡ ♥❡❡❞ t❤✐s ♦r❞❡r ❛s ❛♥ ❛❞❞✐t✐♦♥❛❧ ♣❛rt ♦❢ ❞❛t❛ < ∼ ❝♦♥t❛✐♥❡❞ ✐♥ < − ✦ ❲❡ ❤❛✈❡ x 4 < ∼ x 3 ❜✉t ♥♦t x 3 < ∼ x 4 ✦ ❙✐♠✐❧❛r❧② y 2 < ∼ y 1 ✳ ✸

❉❛t❛ ❢♦r ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ❢❛❝❡s ✿ { S n } n ∈ ω ❀ S n ✐s ❛ ✜♥✐t❡ s❡t ♦❢ ❢❛❝❡s ♦❢ ❞✐♠❡♥s✐♦♥ n ❀ ❛❧♠♦st ❛❧❧ S n ✬s ❛r❡ ❡♠♣t②❀ ❞♦♠❛✐♥ r❡❧❛t✐♦♥ δ ✿ δ ( α ) ✐s ❡✐t❤❡r ❛ ✜♥✐t❡ ♥♦♥✲ ❡♠♣t② s❡t ♦❢ ❢❛❝❡s ♦r ❛♥ ❡♠♣t② ❢❛❝❡s❀ ❇❡❧♦✇ ✇❡ ❤❛✈❡✿ δ ( a 2 ) = 1 s 0 ❛♥❞ δ ( a 1 ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 8 , x 9 } ❝♦❞♦♠❛✐♥ ❢✉♥❝t✐♦♥ γ ✿ ❡✳❣✳ γ ( a 1 ) = x 0 ❧♦✇❡r ♦r❞❡r < ∼ ✿ ❡✳❣✳ x 4 < ∼ x 3 x 5 s 3 s 1 ✲ ✁ ✻ ✒ � ❑ ❆ ❅ ✁ ❆ � ❅ ✁ ❆ � ❅ ⇓ ⇓ ✁ ❆ � ❅ x 8 ✁ ❆ x 2 ✖✕ a 4 ✖✕ a 3 x 3 � ❅ ⇓ a 1 � ❅ x 4 ✔ ✗ � ❅ � ✟ ❍❍❍ ❅ ✟ a 6 x 9 a 2 � ✟ ✕ ✖ ❘ ❅ ✟ ✙ ❍ ❥ ⇒ ⇐ x 1 s 4 s 0 ✲ x 0 ❚❤❡ ✉♣♣❡r ♦r❞❡r < + ✐s ❞❡✜♥❛❜❧❡ ❢r♦♠ γ ❛♥❞ δ ✳ ✹

❆①✐♦♠s ❢♦r ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s ■♥ t❤❡ ♦r❞❡r❡❞ ❢❛❝❡s str✉❝t✉r❡ s 2 s 2 ✒ � ❅ ✒ � ❅ x 5 x 4 x 5 x 4 � � ❅ ❅ ⇓ a 3 α � ❅ ❘ � ❅ ❘ = ⇒ s 3 s 1 s 3 s 1 ✲ x 3 ✏ ✶ ✏✏✏✏✏✏✏✏✏✏✏✏ ⇓ a 0 ⇓ a 2 ✒ � ❅ ✒ � ❅ x 6 x 1 x 6 x 1 x 2 � � ❅ ❅ ⇓ a 1 � ❘ ❅ � ❘ ❅ s 4 s 0 s 4 s 0 ✲ ✲ x 0 x 0 ✇❡ ❤❛✈❡ γγ ( α ) = x 0 ✱ δδ ( α ) = { x 1 , x 2 , x 3 , x 4 , x 5 , x 6 } δγ ( α ) = { x 1 , x 4 , x 5 , x 6 } ✱ γδ ( α ) = { x 0 , x 2 , x 3 } ●❧♦❜✉❧❛r✐t② ❛①✐♦♠ ✭♣♦s✐t✐✈❡ ❝❛s❡✮ γγ ( α ) = γδ ( α ) − δδ ( α ) ✱ δγ ( α ) = δδ ( α ) − γδ ( α ) ✺

❇✉t ✇❤❡♥ ✇❡ ❤❛✈❡ ❧♦♦♣s ✐♥ t❤❡ ❞♦♠❛✐♥ ❛s ✐♥ b 1 ♦r ❡♠♣t②✲❞♦♠❛✐♥ ❧♦♦♣ ❛s ✐♥ b 2 s s ✁ ❑ ❆ ✁ ❑ ❆ ✁ ✁ ✻ ❆ ✁ ❆ ❑ ❆ ✁ ❆ β ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ ✁ ❆ = ⇒ ⇓ ⇓ ✁ ❆ ✁ ❆ ✁ ❆ ⇓ b 0 ✁ ❆ ✖✕ ✖✕ b 3 b 2 ✁ ❆ ✁ ❆ y 2 y 1 ⇓ b 1 y 0 y 0 ✫ ✪ ✫ ✪ ✇❡ ❤❛✈❡ γγ ( b i ) = δδ ( b i ) = δγ ( b i ) = γδ ( b i ) = s ✱ ❛♥❞ t❤❡ ❛❜♦✈❡ ❢♦r♠✉❧❛s ❞♦❡s ♥♦t ✇♦r❦✳ ❲❡ ❤❛✈❡ t♦ ❞r♦♣ ❜♦t❤ ❧♦♦♣s ❛♥❞ ❡♠♣t② ❢❛❝❡s✳ ●❧♦❜✉❧❛r✐t② ❛①✐♦♠ δ − λ ( α ) γγ ( α ) = γδ ( α ) − δ ˙ δ − λ ( α ) δγ ( α ) ≡ 1 δδ ( α ) − γ ˙ ≡ 1 ✐s ✬❡q✉❛❧✐t②✬ t❤❛t ✐❣♥♦r❡s ❡♠♣t② ❢❛❝❡s✱ ✐✳❡✳ t❤❡ ❡♠♣t② ❢❛❝❡s t❤❛t ♠✐❣❤t ♦❝❝✉r ♦♥ t❤❡ r✐❣❤t s✐❞❡ ♦❢ t❤❡ s✐❣♥ ≡ 1 ♠✉st ❜❡ ❡♠♣t② ♦♥ ❡t❤❡r ❞♦♠❛✐♥ ♦r ❝♦❞♦♠❛✐♥ ♦❢ ❛ ❢❛❝❡ t❤❛t ❜❡❧♦♥❣s t♦ t❤❡ ❧❡❢t s✐❞❡✳ ✻

❖t❤❡r ❛①✐♦♠s ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s t❛❧❦s ❛❜♦✉t t❤❡ ✉♣♣❡r < + ❛♥❞ ❧♦✇❡r < ∼ ♦r❞❡rs✳ ❚❤❡② ❛r❡ str✐❝t✱ ❞✐s❥♦✐♥t ❛♥❞ < ∼ ✐s ♠❛①✐♠❛❧ s✉❝❤ ❝♦♥t❛✐♥❡❞ ✐♥ < − ✳ ❚❤❡ ✉♣♣❡r ♦r❞❡r ♦♥ 0 ✲❝❡❧❧s ✐s ❧✐♥❡❛r✳ ◆♦ t✇♦ ❢❛❝❡s ✐♥ ❛ ❞♦♠❛✐♥ ♦❢ ❛ ❢❛❝❡ ♠✐❣❤t ❜❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ t❤❡ ✉♣♣❡r ♦r❞❡r < + ✳ ■♥❝✐❞❡♥t ❢❛❝❡s ♠✉st ❜❡ ❝♦♠♣❛r❛❜❧❡ ✐♥ ♦♥❡ ♦❢ t❤❡s❡ ♦r❞❡rs✳ ❊✈❡r② ❧♦♦♣ ♠✉st ❜❡ ✜❧❧❡❞ ✐♥✱ ✐✳❡✳ ♠✉st ❜❡ ❛ ❝♦❞♦♠❛✐♥ ♦❢ ❛ ❝❡❧❧ ✇❤✐❝❤ ✐s ♥♦t ❛ ❧♦♦♣✳ ✼

❚❤❡r❡ ❛r❡ t✇♦ ❜❛s✐❝ ❦✐♥❞s ♦❢ ♠♦r♣❤✐s♠s ♦❢ ♦r✲ ❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s✳ ❆ ❧♦❝❛❧ ♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s f : S → T ✐s ❛ ❢❛♠✐❧② ♦❢ ❢✉♥❝t✐♦♥s f k : S k → T k ✱ ❢♦r k ∈ ω ✱ s✉❝❤ t❤❛t t❤❡ ❞✐❛❣r❛♠s f k +1 f k +1 S k +1 T k +1 S k +1 T k +1 ✲ ✲ γ γ δ ❄ δ ❄ ❄ ❄ S k ⊔ · 1 S k − 1 T k ⊔ · 1 T k − 1 S k T k ✲ ✲ f k + 1 f k − 1 f k ❝♦♠♠✉t❡✳ ❋♦r t❤❡ r✐❣❤t sq✉❛r❡ ✐t ♠❡❛♥s ♠♦r❡ t❤❡♥ ❝♦♠♠✉t❛t✐♦♥ ♦❢ r❡❧❛t✐♦♥s✱ ✇❡ ❞❡♠❛♥❞ t❤❛t ❢♦r ❛♥② a ∈ S ≥ 1 ✱ δ ( a ) , < ∼ ) − δ ( f ( a )) , < ∼ ) f a : (˙ → (˙ ❜❡ ❛♥ ♦r❞❡r ✐s♦♠♦r♣❤✐s♠✱ ✇❤❡r❡ f a ✐s t❤❡ r❡✲ str✐❝t✐♦♥ ♦❢ f t♦ ˙ δ ( a ) ✭✐❢ δ ( a ) = 1 u ✇❡ ♠❡❛♥ ❜② t❤❛t δ ( f ( a )) = 1 f ( u ) ✮✳ ❆ ❣❧♦❜❛❧ ✭♠♦♥♦t♦♥❡✮ ♠♦r♣❤✐s♠ ♦❢ ♦r❞❡r❡❞ ❢❛❝❡ str✉❝t✉r❡s f : S → T ✐s ❛ ❧♦❝❛❧ ♠♦r♣❤✐s♠ t❤❛t ♣r❡s❡r✈❡s ❧♦✇❡r ♦r❞❡r < ∼ ✭❣❧♦❜❛❧❧②✮✳ ✽