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❆ ❘❛♠s❡②✲❚❤❡♦r❡t✐❝ ◆♦t✐♦♥ ♦❢ ❋♦r❝✐♥❣ ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❙❡t❚♦♣ ◆♦✈✐ ❙❛❞ ❏✉❧② ✸✱ ✷✵✶✽

✭✷✮ ✳ ✭✸✮ ❋♦r ✱ ✇❡ ❧❡t ❞❡♥♦t❡ ✱ ✐✳❡✳✱ ✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② ✱ ✳ ❚❤❡ s❡t ✐s t❤❡ s❡t ♦❢ ✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳ ❚❤❡ k ✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fin k ❉❡✜♥✐t✐♦♥ ▲❡t k ∈ ω \ { 0 } ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a : ω → k + 1 ✇❡ ❧❡t supp( a ) = { n ∈ ω : a ( n ) � = 0 } ✳ Fin k = { a : ω → k + 1 : supp( a ) ✐s ✜♥✐t❡ ∧ k ∈ range( a ) } .

✭✸✮ ❋♦r ✱ ✇❡ ❧❡t ❞❡♥♦t❡ ✱ ✐✳❡✳✱ ✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ ♦❢ ❡❧❡♠❡♥ts ♦❢ ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② ✱ ✳ ❚❤❡ s❡t ✐s t❤❡ s❡t ♦❢ ✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳ ❚❤❡ k ✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fin k ❉❡✜♥✐t✐♦♥ ▲❡t k ∈ ω \ { 0 } ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a : ω → k + 1 ✇❡ ❧❡t supp( a ) = { n ∈ ω : a ( n ) � = 0 } ✳ Fin k = { a : ω → k + 1 : supp( a ) ✐s ✜♥✐t❡ ∧ k ∈ range( a ) } . ✭✷✮ Fin [1 ,k ] = � k j =1 Fin j ✳

❚❤❡ k ✲✈❛❧✉❡❞ ❜❧♦❝❦s✱ Fin k ❉❡✜♥✐t✐♦♥ ▲❡t k ∈ ω \ { 0 } ✉♥❧❡ss st❛t❡❞ ♦t❤❡r✇✐s❡✳ ✭✶✮ ❋♦r a : ω → k + 1 ✇❡ ❧❡t supp( a ) = { n ∈ ω : a ( n ) � = 0 } ✳ Fin k = { a : ω → k + 1 : supp( a ) ✐s ✜♥✐t❡ ∧ k ∈ range( a ) } . ✭✷✮ Fin [1 ,k ] = � k j =1 Fin j ✳ ✭✸✮ ❋♦r a, b ∈ Fin k ✱ ✇❡ ❧❡t a < b ❞❡♥♦t❡ supp( a ) < supp( b ) ✱ ✐✳❡✳✱ ( ∀ m ∈ supp( a ))( ∀ n ∈ supp( b ))( m < n ) ✳ ❆ ✜♥✐t❡ ♦r ✐♥✜♥✐t❡ s❡q✉❡♥❝❡ � a i : i < m ≤ ω � ♦❢ ❡❧❡♠❡♥ts ♦❢ Fin k ✐s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥ ✐❢ ❢♦r ❛♥② i < j < m ✱ a i < a j ✳ ❚❤❡ s❡t (Fin k ) ω ✐s t❤❡ s❡t ♦❢ ω ✲s❡q✉❡♥❝❡s ✐♥ ❜❧♦❝❦✲♣♦s✐t✐♦♥✱ ❛❧s♦ ❝❛❧❧❡❞ ❜❧♦❝❦ s❡q✉❡♥❝❡s✳

✭✺✮ ❋♦r ❛♥② ✇❡ ❞❡✜♥❡ ♦♥ t❤❡ ❚❡tr✐s ♦♣❡r❛t✐♦♥✿ ❜② ✳ ❚✇♦ ♦♣❡r❛t✐♦♥s ♦♥ Fin j ❉❡✜♥✐t✐♦♥ ✭✹✮ ❋♦r k ≥ 1 ✱ a, b ∈ Fin k ✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣❛rt✐❛❧ s❡♠✐❣r♦✉♣ ♦♣❡r❛t✐♦♥ + ❛s ❢♦❧❧♦✇s✿ ■❢ supp( a ) < supp( b ) ✱ t❤❡♥ a + b ∈ Fin k ✐s ❞❡✜♥❡❞✳ ❲❡ ❧❡t ( a + b )( n ) = a ( n ) + b ( n ) ✳ ❖t❤❡r✇✐s❡ a + b ✐s ✉♥❞❡✜♥❡❞✳ ❚❤✉s a + b = a ↾ supp( a ) ∪ b ↾ supp( b ) ∪ 0 ↾ ( ω \ (supp( a ) ∪ supp( b ))) ✳

❚✇♦ ♦♣❡r❛t✐♦♥s ♦♥ Fin j ❉❡✜♥✐t✐♦♥ ✭✹✮ ❋♦r k ≥ 1 ✱ a, b ∈ Fin k ✱ ✇❡ ❞❡✜♥❡ t❤❡ ♣❛rt✐❛❧ s❡♠✐❣r♦✉♣ ♦♣❡r❛t✐♦♥ + ❛s ❢♦❧❧♦✇s✿ ■❢ supp( a ) < supp( b ) ✱ t❤❡♥ a + b ∈ Fin k ✐s ❞❡✜♥❡❞✳ ❲❡ ❧❡t ( a + b )( n ) = a ( n ) + b ( n ) ✳ ❖t❤❡r✇✐s❡ a + b ✐s ✉♥❞❡✜♥❡❞✳ ❚❤✉s a + b = a ↾ supp( a ) ∪ b ↾ supp( b ) ∪ 0 ↾ ( ω \ (supp( a ) ∪ supp( b ))) ✳ ✭✺✮ ❋♦r ❛♥② k ≥ 2 ✇❡ ❞❡✜♥❡ ♦♥ Fin k t❤❡ ❚❡tr✐s ♦♣❡r❛t✐♦♥✿ T : Fin k → Fin k − 1 ❜② T ( a )( n ) = max { a ( n ) − 1 , 0 } ✳

●❡♥❡r❛t❡❞ s❡♠✐❣r♦✉♣s ❉❡✜♥✐t✐♦♥ ✭✻✮ ▲❡t B ⊆ Fin k ❜❡ ♠✐♥✲✉♥❜♦✉♥❞❡❞✱ ✐✳❡✳✱ ❝♦♥t❛✐♥ ❢♦r ❛♥② n s♦♠❡ a ✇✐t❤ supp( a ) > n ✳ ❲❡ ❧❡t TFU k ( B ) = { T ( j 0 ) ( b n 0 ) + · · · + T ( j ℓ ) ( b n ℓ ) : ℓ ∈ ω \ { 0 } , b n i ∈ B, b n 0 < · · · < b n ℓ , j i ∈ k, ∃ r ≤ ℓj r = 0 } ❜❡ t❤❡ ♣❛rt✐❛❧ s✉❜s❡♠✐❣r♦✉♣ ♦❢ Fin k ❣❡♥❡r❛t❡❞ ❜② B ✳ ❲❡ ❝❛❧❧ B ❛ TFU k ✲s❡t ✐❢ B = TFU k ( B ) ✳

✭✽✮ ❲❡ ❞❡✜♥❡ t❤❡ ✲♦♣❡r❛t✐♦♥✿ ▲❡t ❛♥❞ ✳ ✇✐t❤ ✳ ❚❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r ❉❡✜♥✐t✐♦♥ a ⊑ k ¯ a ∈ (TFU k (¯ b )) ω ✳ ✭✼✮ ❲❡ ❞❡✜♥❡ t❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r✿ ¯ b ✐❢ ¯

❚❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r ❉❡✜♥✐t✐♦♥ a ⊑ k ¯ a ∈ (TFU k (¯ b )) ω ✳ ✭✼✮ ❲❡ ❞❡✜♥❡ t❤❡ ❝♦♥❞❡♥s❛t✐♦♥ ♦r❞❡r✿ ¯ b ✐❢ ¯ a ∈ (Fin k ) ω ❛♥❞ s ∈ Fin k ✳ ✭✽✮ ❲❡ ❞❡✜♥❡ t❤❡ past ✲♦♣❡r❛t✐♦♥✿ ▲❡t ¯ (¯ a past s ) = � a i : i ≥ i 0 � ✇✐t❤ i 0 = min { i : supp( a i ) > supp( s ) } ✳

✷✳ ▲❡t ❛♥❞ ❧❡t ❜❡ ❛ ✲♣♦✐♥t✳ ❲❡ s❛② ❛✈♦✐❞s ✐❢ ✐s ♥♥❝ t♦ ✳ ❆ ♥❡❣❛t✐♦♥ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❢♦r ♥♦t ♥❡❝❡ss❛r✐❧② ❝❡♥tr❡❞ ❢❛♠✐❧✐❡s ❉❡✜♥✐t✐♦♥ ✶✳ ❚✇♦ s✉❜s❡ts F 1 ✱ F 2 ♦❢ [ ω ] ω ❛r❡ ❝❛❧❧❡❞ ♥♥❝✱ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ ❢♦r ❛♥② X i ∈ F i ✱ i = 1 , 2 ❛♥❞ ❛♥② ✜♥✐t❡✲t♦✲♦♥❡ h : ω → ω t❤❡r❡ ✐s Y i ⊆ X i ✱ Y i ∈ F i ✱ i = 1 , 2 s✉❝❤ t❤❛t h [ Y 1 ] ∩ h [ Y 2 ] = ∅ ✳

❆ ♥❡❣❛t✐♦♥ ♦❢ ♥❡❛r ❝♦❤❡r❡♥❝❡ ❢♦r ♥♦t ♥❡❝❡ss❛r✐❧② ❝❡♥tr❡❞ ❢❛♠✐❧✐❡s ❉❡✜♥✐t✐♦♥ ✶✳ ❚✇♦ s✉❜s❡ts F 1 ✱ F 2 ♦❢ [ ω ] ω ❛r❡ ❝❛❧❧❡❞ ♥♥❝✱ ♥♦t ♥❡❛r❧② ❝♦❤❡r❡♥t✱ ✐❢ ❢♦r ❛♥② X i ∈ F i ✱ i = 1 , 2 ❛♥❞ ❛♥② ✜♥✐t❡✲t♦✲♦♥❡ h : ω → ω t❤❡r❡ ✐s Y i ⊆ X i ✱ Y i ∈ F i ✱ i = 1 , 2 s✉❝❤ t❤❛t h [ Y 1 ] ∩ h [ Y 2 ] = ∅ ✳ ✷✳ ▲❡t H ⊆ (Fin k ) ω ❛♥❞ ❧❡t E ❜❡ ❛ P ✲♣♦✐♥t✳ ❲❡ s❛② H ❛✈♦✐❞s E ✐❢ { supp(¯ a ) : ¯ a ∈ H} ✐s ♥♥❝ t♦ E ✳

❆ s✉❜s♣❛❝❡ ♦❢ (Fin k ) ω ✕❋✐①✐♥❣ PP ❛♥❞ ¯ R ❉❡✜♥✐t✐♦♥ ❲❡ ✜① ♣❛r❛♠❡t❡rs ❛s ❢♦❧❧♦✇s✳ ▲❡t k ≥ 1 ✳ ❋✐① P min , P max ⊆ { 1 , . . . , k } ✳ ▲❡t PP = { ( i, x ) : x ∈ { min , max } , i ∈ P x } ❛♥❞ ❧❡t ¯ R = { ( ι, R ι ) : ι ∈ PP } ❜❡ ❛ PP ✲s❡q✉❡♥❝❡ ♦❢ ♣❛✐r✇✐s❡ ♥♥❝ ❘❛♠s❡② ✉❧tr❛✜❧t❡rs ✭♣❛✐r✇✐s❡ ♥♥❝ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s✱ ✐✳❡✳✱ ❤❛♣♣② ❢❛♠✐❧✐❡s✱ ✇♦✉❧❞ s✉✣❝❡ ❢♦r t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt② ❛♥❞ ♣r♦♣❡r♥❡ss✮✳ ❲❡ ❛❧s♦ ♥❛♠❡ t❤❡ ❡♥❞ s❡❣♠❡♥ts ❢♦r 1 ≤ j ≤ k ✿ ¯ R ↾ { j, . . . , k } = { ( ι, R ι ) : ι = ( i, x ) ∈ PP ∧ i ∈ { j, . . . , k }} .

❆ ❘❛♠s❡② ✉❧tr❛✜❧t❡r ✐s ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ t❤❛t ✐s ❛❧s♦ ❛ ✜❧t❡r✳ ❍❛♣♣② ❢❛♠✐❧✐❡s ✕ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s ❉❡✜♥✐t✐♦♥ ❲❡ ❝❛❧❧ H ⊆ [ ω ] ω ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ ✐❢ ✶✳ ❛♥② ❝♦✜♥✐t❡ s✉❜s❡t ♦❢ ω ✐s ✐♥ H ✱ ✷✳ ∀ X ∈ H∀ X 1 , X 2 ( X 1 ∪ X 2 = X → X 1 ∈ H ∨ X 2 ∈ H ) ✳ ✸✳ ❋♦r ❛♥② � A n : n < ω � s✉❝❤ t❤❛t ❢♦r ❛♥② n ✱ A n ∈ H ❛♥❞ A n +1 ⊆ A n t❤❡r❡ ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ A ∈ H ✱ ✐✳❡✳✱ ∀ n ∈ A ( A \ ( n + 1) ⊆ A n ) .

❍❛♣♣② ❢❛♠✐❧✐❡s ✕ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧s ❉❡✜♥✐t✐♦♥ ❲❡ ❝❛❧❧ H ⊆ [ ω ] ω ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ ✐❢ ✶✳ ❛♥② ❝♦✜♥✐t❡ s✉❜s❡t ♦❢ ω ✐s ✐♥ H ✱ ✷✳ ∀ X ∈ H∀ X 1 , X 2 ( X 1 ∪ X 2 = X → X 1 ∈ H ∨ X 2 ∈ H ) ✳ ✸✳ ❋♦r ❛♥② � A n : n < ω � s✉❝❤ t❤❛t ❢♦r ❛♥② n ✱ A n ∈ H ❛♥❞ A n +1 ⊆ A n t❤❡r❡ ✐s ❛ ❞✐❛❣♦♥❛❧ ❧♦✇❡r ❜♦✉♥❞ A ∈ H ✱ ✐✳❡✳✱ ∀ n ∈ A ( A \ ( n + 1) ⊆ A n ) . ❆ ❘❛♠s❡② ✉❧tr❛✜❧t❡r ✐s ❛ s❡❧❡❝t✐✈❡ ❝♦✐❞❡❛❧ t❤❛t ✐s ❛❧s♦ ❛ ✜❧t❡r✳