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▼✐❧❧❡r ❋♦r❝✐♥❣ ✇✐t❤ ❈❛♥❥❛r ❯❧tr❛✜❧t❡rs ❍❡✐❦❡ ▼✐❧❞❡♥❜❡r❣❡r ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❈❤r✐st✐❛♥ ❇rä✉♥✐♥❣❡r ❍❛♠❜✉r❣ ❙❡t ❚❤❡♦r② ❲♦r❦s❤♦♣ ♦♥❧✐♥❡ ❈♦♥❢❡r❡♥❝❡ ❤♦st❡❞ ❜② ❍❛♠❜✉r❣ ❏✉♥❡ ✷✵ ✕ ✷✶✱ ✷✵✷✵ ✶ ✴ ✸✵

❋♦r ✇❡ ❤❛✈❡ ❛♥❞ ✈✐❝❡ ✈❡rs❛✱ ❣♦✐♥❣ ❢r♦♠ ✜♥✐t❡ s✉❜s❡ts ♦❢ t♦ t❤❡✐r ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ❋✐♥✐t❡ ❙tr✐❝t❧② ■♥❝r❡❛s✐♥❣ ❙❡q✉❡♥❝❡s ✕ ❇❧♦❝❦s ❉❡✜♥✐t✐♦♥ ❚❤❡ s❡t ♦❢ ✜♥✐t❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ω ↑ <ω ✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ s ∈ ω ↑ <ω ✱ | s | ✱ ✐s ✐ts ❞♦♠❛✐♥✳ ❋♦r s, t ∈ ω <ω ✱ ✇❡ s❛② ✏ t ❡①t❡♥❞s s ✑ ♦r ✏ s ✐s ❛♥ ✐♥✐t✐❛❧ s❡❣♠❡♥t ♦❢ t ✑ ❛♥❞ ✇r✐t❡ s � t ✐❢ dom( s ) ⊆ dom( t ) ❛♥❞ s = t ↾ dom( s ) ✳ ✷ ✴ ✸✵

❋✐♥✐t❡ ❙tr✐❝t❧② ■♥❝r❡❛s✐♥❣ ❙❡q✉❡♥❝❡s ✕ ❇❧♦❝❦s ❉❡✜♥✐t✐♦♥ ❚❤❡ s❡t ♦❢ ✜♥✐t❡ str✐❝t❧② ✐♥❝r❡❛s✐♥❣ s❡q✉❡♥❝❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs ✐s ❝❛❧❧❡❞ ω ↑ <ω ✳ ❚❤❡ ❧❡♥❣t❤ ♦❢ s ∈ ω ↑ <ω ✱ | s | ✱ ✐s ✐ts ❞♦♠❛✐♥✳ ❋♦r s, t ∈ ω <ω ✱ ✇❡ s❛② ✏ t ❡①t❡♥❞s s ✑ ♦r ✏ s ✐s ❛♥ ✐♥✐t✐❛❧ s❡❣♠❡♥t ♦❢ t ✑ ❛♥❞ ✇r✐t❡ s � t ✐❢ dom( s ) ⊆ dom( t ) ❛♥❞ s = t ↾ dom( s ) ✳ ❋♦r s, t ∈ ω ↑ <ω ✇❡ ❤❛✈❡ s � t ⇔ range( s ) ⊑ range( t ) ❛♥❞ ✈✐❝❡ ✈❡rs❛✱ ❣♦✐♥❣ ❢r♦♠ ✜♥✐t❡ s✉❜s❡ts ♦❢ ω t♦ t❤❡✐r ✐♥❝r❡❛s✐♥❣ ❡♥✉♠❡r❛t✐♦♥s✳ ✷ ✴ ✸✵

❚r❡❡s ❉❡✜♥✐t✐♦♥ ❆ s✉❜s❡t p ⊆ ω ↑ <ω t❤❛t ✐s ❝❧♦s❡❞ ✉♥❞❡r ✐♥✐t✐❛❧ s❡❣♠❡♥ts ✐s ❝❛❧❧❡❞ ❛ tr❡❡✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ ❛ tr❡❡ ❛r❡ ❝❛❧❧❡❞ ♥♦❞❡s✳ ❆ ♥♦❞❡ s ∈ p ✐s ❝❛❧❧❡❞ ❛ s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ✐❢ s ❤❛s ♠♦r❡ t❤❛♥ ♦♥❡ ❞✐r❡❝t ⊳ ✲s✉❝❝❡ss♦r ✐♥ p ❛♥❞ ω ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ✐❢ s ❤❛s ✐♥✜♥✐t❡❧② ♠❛♥② ❞✐r❡❝t ⊳ ✲s✉❝❝❡ss♦rs ✐♥ p ✳ ❚❤❡ s❡t ♦❢ s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p ✐s ❞❡♥♦t❡❞ ❜② sp( p ) ✇❤✐❧❡ ω ✲ sp( p ) ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ω ✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p ✳ ✸ ✴ ✸✵

❙♣❡❝✐❛❧ ❙❡ts ♦❢ ❇❧♦❝❦s ❉❡✜♥✐t✐♦♥ ✭✶✮ ❋♦r ❛♥② s❡t A ✇❡ ✇r✐t❡ [ A ] <ω = { t : t ⊆ A, | t | < ω } ✳ ❚❤❡ ❡❧❡♠❡♥ts ♦❢ Fin = [ ω ] <ω \ {∅} ❛r❡ ❝❛❧❧❡❞ ❜❧♦❝❦s ✳ ✭✷✮ ▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω ✳ ❲❡ ❧❡t F <ω = { [ A ] <ω \ {∅} : A ∈ F} ( F <ω ) + = { B ⊆ Fin : ∀ A ∈ F ([ A ] <ω ∩ B � = ∅ ) } ✹ ✴ ✸✵

❙✉❝❤ ❛ ✐s ❝❛❧❧❡❞ ❛♥ ✲s♣❧✐tt✐♥❣ ♥♦❞❡✳ ❲❡ ❢✉rt❤❡r♠♦r❡ r❡q✉✐r❡ ♦❢ t❤❛t ❡❛❝❤ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ✐s ❛♥ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❛♥❞ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ✲♠✐♥✐♠❛❧ ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❝❛❧❧❡❞ t❤❡ tr✉♥❦ ♦❢ ✱ ✳ ❚❤❡ s❡t ♦❢ ✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ ✐s ❞❡♥♦t❡❞ ❜② ✲ ✳ ❋❛♠✐❧✐❡s ♦❢ ❙✉♣❡r♣❡r❢❡❝t ❚r❡❡s ●✉③♠á♥ ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐ ✐♥tr♦❞✉❝❡❞ ❛ ❢❛♠✐❧② ♦❢ ▼✐❧❧❡r ❢♦r❝✐♥❣s PT ( F ) ✱ F ❛ ✜❧t❡r ♦✈❡r ω ✱ ❡①t❡♥❞✐♥❣ t❤❡ ❋ré❝❤❡t ✜❧t❡r✳ ❉❡✜♥✐t✐♦♥ ▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω ✳ ❚❤❡ ❢♦r❝✐♥❣ PT ( F ) ❝♦♥s✐sts ♦❢ ❛❧❧ p ⊆ ω ↑ <ω s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ s ∈ p t❤❡r❡ ✐s t � s ✱ s✉❝❤ t❤❛t t ∈ ω ✲ sp( p ) ❛♥❞ sucspl p ( t ) := { range( r ) \ range( t ) : r ❛ ⊳ ✲♠✐♥✐♠❛❧ ✐♥✜♥✐t❡❧② s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ❛❜♦✈❡ t } ∈ ( F <ω ) + . ✺ ✴ ✸✵

❋❛♠✐❧✐❡s ♦❢ ❙✉♣❡r♣❡r❢❡❝t ❚r❡❡s ●✉③♠á♥ ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐ ✐♥tr♦❞✉❝❡❞ ❛ ❢❛♠✐❧② ♦❢ ▼✐❧❧❡r ❢♦r❝✐♥❣s PT ( F ) ✱ F ❛ ✜❧t❡r ♦✈❡r ω ✱ ❡①t❡♥❞✐♥❣ t❤❡ ❋ré❝❤❡t ✜❧t❡r✳ ❉❡✜♥✐t✐♦♥ ▲❡t F ❜❡ ❛ ✜❧t❡r ♦✈❡r ω ✳ ❚❤❡ ❢♦r❝✐♥❣ PT ( F ) ❝♦♥s✐sts ♦❢ ❛❧❧ p ⊆ ω ↑ <ω s✉❝❤ t❤❛t ❢♦r ❡❛❝❤ s ∈ p t❤❡r❡ ✐s t � s ✱ s✉❝❤ t❤❛t t ∈ ω ✲ sp( p ) ❛♥❞ sucspl p ( t ) := { range( r ) \ range( t ) : r ❛ ⊳ ✲♠✐♥✐♠❛❧ ✐♥✜♥✐t❡❧② s♣❧✐tt✐♥❣ ♥♦❞❡ ♦❢ p ❛❜♦✈❡ t } ∈ ( F <ω ) + . ❙✉❝❤ ❛ t ✐s ❝❛❧❧❡❞ ❛♥ F ✲s♣❧✐tt✐♥❣ ♥♦❞❡✳ ❲❡ ❢✉rt❤❡r♠♦r❡ r❡q✉✐r❡ ♦❢ p t❤❛t ❡❛❝❤ ω ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ✐s ❛♥ F ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❛♥❞ t❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ⊳ ✲♠✐♥✐♠❛❧ ω ✲s♣❧✐tt✐♥❣ ♥♦❞❡ ❝❛❧❧❡❞ t❤❡ tr✉♥❦ ♦❢ p ✱ tr( p ) ✳ ❚❤❡ s❡t ♦❢ F ✲s♣❧✐tt✐♥❣ ♥♦❞❡s ♦❢ p ✐s ❞❡♥♦t❡❞ ❜② F ✲ sp( p ) ✳ ✺ ✴ ✸✵

P❧❛✐♥ s❧✐❞❡ ❢♦r ❛ s❦❡t❝❤ ✻ ✴ ✸✵

●✉③♠á♥✬s ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐✬s ❘❡s✉❧ts ▲❡♠♠❛ ❚❤❡ ❢♦r❝✐♥❣ PT ( F ) ❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✳ p ∈ PT ( F ) ✱ s ∈ F ✲ sp( p ) ✱ D ⊆ PT ( F ) ♦♣❡♥ ❞❡♥s❡✳ ❚❤❡♥ E ( p, s, D ) = ⊑ − min { range( t ) \ range( s ) : ∃ q ≤ p (tr( q ) = t ∧ q ∈ D ) } ✐s ✐♥ ( F <ω ) + ✳ ▲❡t f ∈ F ✳ p ↾↾ F ❝♦♥t❛✐♥s ♦♥❧② t❤♦s❡ ♥♦❞❡s t ∈ p ❢♦r ✇❤✐❝❤ range( t ) \ tr( p ) ⊆ F ✳ ❲❡ ❤❛✈❡ p ↾↾ F ≤ 0 p ✳ ✼ ✴ ✸✵

●✉③♠á♥✬s ❛♥❞ ❑❛❧❛❥❞③✐❡✈s❦✐✬s ❘❡s✉❧ts ▲❡♠♠❛ ❚❤❡ ❢♦r❝✐♥❣ PT ( F ) ❤❛s t❤❡ ♣✉r❡ ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✳ p ∈ PT ( F ) ✱ s ∈ F ✲ sp( p ) ✱ D ⊆ PT ( F ) ♦♣❡♥ ❞❡♥s❡✳ ❚❤❡♥ E ( p, s, D ) = ⊑ − min { range( t ) \ range( s ) : ∃ q ≤ p (tr( q ) = t ∧ q ∈ D ) } ✐s ✐♥ ( F <ω ) + ✳ ▲❡t f ∈ F ✳ p ↾↾ F ❝♦♥t❛✐♥s ♦♥❧② t❤♦s❡ ♥♦❞❡s t ∈ p ❢♦r ✇❤✐❝❤ range( t ) \ tr( p ) ⊆ F ✳ ❲❡ ❤❛✈❡ p ↾↾ F ≤ 0 p ✳ ✼ ✴ ✸✵

❋♦r❝✐♥❣ ✇✐t❤ F σ ✲❋✐❧t❡rs ❉❡✜♥✐t✐♦♥ ✭✶✮ ❚❤❡ ♣❛rt✐❛❧ ♦r❞❡r F σ ✐s t❤❡ ❢♦r❝✐♥❣ ✇✐t❤ F σ ✲✜❧t❡rs ♦✈❡r ω ✳ ❙tr♦♥❣❡r ✜❧t❡rs ❛r❡ s✉♣❡r✜❧t❡rs✳ ✭✷✮ ■❢ F ✐s ❛ ✜❧t❡r✱ t❤❡♥ F σ ( F ) ✐s t❤❡ ❢♦r❝✐♥❣ ✇✐t❤ F σ ✲✜❧t❡rs t❤❛t ❛r❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ F ✱ ✐✳❡✳ G ∈ F σ ( F ) ✐✛ G ✐s ❛♥ F σ ✲✜❧t❡r ❛♥❞ G ⊆ F + = { X ⊆ ω : ∀ ( F ∈ F )( X ∩ F � = ∅ ) } ✳ ❉❡✜♥✐t✐♦♥ ▲❡t G ❜❡ ❛♥ F σ ( F ) ✲❣❡♥❡r✐❝ ✜❧t❡r✳ ❲❡ ❧❡t U ❜❡ ❛ F σ ( F ) ✲♥❛♠❡ ❢♦r t❤❡ ✉♥✐♦♥ ♦❢ G ✳ ❇② ❛ ❞❡♥s✐t② ❛r❣✉♠❡♥t✱ t❤❡ ♣♦s❡t F σ ( F ) ❢♦r❝❡s t❤❛t U ✐s ❛♥ ✉❧tr❛✜❧t❡r t❤❛t ❝♦♥t❛✐♥s F ❛s ❛ s✉❜s❡t✳ ✽ ✴ ✸✵