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❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❙✐❡❞❧❝❡ ❯♥✐✈❡rs✐t② ♦❢ ◆❛t✉r❛❧ ❙❝✐❡♥❝❡s ❛♥❞ ❍✉♠❛♥✐t✐❡s ✭ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ❏❛♥ ❑r❡♠♣❛✮ ❆❆❆✽✽ ❲♦r❦s❤♦♣ ♦♥ ●❡♥❡r❛❧ ❆❧❣❡❜r❛ ❲❛rs❛✇ ❯♥✐✈❡rs✐t② ♦❢ ❚❡❝❤♥♦❧♦❣② ❏✉♥❡ ✷✵✲✷✷✱ ✷✵✶✹ ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

■♥ t❤✐s ♥♦t❡s K ✇✐❧❧ ❜❡ ❛ ✜❡❧❞✳ ❉❡✜♥✐t✐♦♥ ❆♥ ❛❧❣❡❜r❛ ♦✈❡r ❛ ✜❡❧❞ K ✐s ❛ ✈❡❝t♦r s♣❛❝❡ ❆ ♦✈❡r K t♦❣❡t❤❡r ✇✐t❤ ❛ ❜✐❧✐♥❡❛r ❛ss♦❝✐❛t✐✈❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ❢♦r ❛r❜✐tr❛r② ❡❧❡♠❡♥ts ❛ , ❜ , ❝ ∈ ❆ ❛♥❞ ❢♦r ❛r❜✐tr❛r② λ ∈ K t❤❡ ❢♦❧❧♦✇✐♥❣ ❡q✉❛❧✐t✐❡s ❛r❡ s❛t✐s✜❡❞✿ ✶ ❛ ( ❜ + ❝ ) = ❛❜ + ❛❝ ❀ ✷ ( ❜ + ❝ ) ❛ = ❜❛ + ❝❛ ❀ ✸ ( ❛❜ ) ❝ = ❛ ( ❜❝ ) ❀ ✹ ( λ ❛ ) ❜ = ❛ ( λ ❜ ) = λ ( ❛❜ ) . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❊①❛♠♣❧❡s ♦❢ ❛❧❣❡❜r❛s ♦✈❡r ❛ ✜❡❧❞✿ ❆♥② ✜❡❧❞ ❡①t❡♥s✐♦♥ L ⊇ K , ❚❤❡ q✉❛t❡r♥✐♦♥s ♦✈❡r r❡❛❧ ♥✉♠❜❡rs✱ ❋♦r ❡✈❡r② ✶ ≤ ♥ < ∞ t❤❡ s❡t ♦❢ ❛❧❧ ♥✲❜②✲♥ ♠❛tr✐❝❡s ♦✈❡r K ✇✐t❤ st❛♥❞❛r❞ ♦♣❡r❛❝t✐♦♥s✱ ❚❤❡ ♥♦♥❝♦♠♠✉t❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ r✐♥❣ K { ① ✶ , . . . , ① ♥ } ✇✐t❤ ✐♥❞❡t❡r♠✐♥❛t❡s ① ✶ , . . . , ① ♥ ❛♥❞ ❝♦❡✣❝✐❡♥ts ✐♥ K . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❆♥ ❛❧❣❡❜r❛ ❆ ✐s s❛✐❞ t♦ ❜❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ♦r ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛❝❝♦r❞✐♥❣ t♦ ✇❤❡t❤❡r t❤❡ s♣❛❝❡ ❆ ✐s ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ♦r ✐♥✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ♦✈❡r K . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❆ K ✲s✉❜s♣❛❝❡ ■ ♦❢ ❛ K ✲❛❧❣❡❜r❛ ❆ ✐s s❛✐❞ t♦ ❜❡ ❛ ❧❡❢t ✐❞❡❛❧ ✐❢ ∀ ❛ ∈ ❆ ∀ ✐ ∈ ■ ❛✐ ∈ ■ ( ❆■ ⊆ ■ ) . ❙✐♠✐❧❛r❧②✱ ❛ K ✲s✉❜s♣❛❝❡ ❏ ⊆ ❆ ✇❤✐❝❤ s❛t✐s✜❡s ∀ ❛ ∈ ❆ ∀ ❥ ∈ ❏ ❥❛ ∈ ❏ ( ❏❆ ⊆ ❏ ) ✐s ❛ r✐❣❤t ✐❞❡❛❧ ✐♥ ❆ . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❆❣r❡❡♠❡♥ts ❆❧❧ ❛❧❣❡❜r❛s ❛r❡ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✱ ✇✐t❤ ✶ � = ✵ . ❆❧❧ ❧❛tt✐❝❡s ❤❛✈❡ t❤❡ s♠❛❧❧❡st ❡❧❡♠❡♥t ω ❛♥❞ t❤❡ ❧❛r❣❡st ❡❧❡♠❡♥t Ω � = ω. ❋♦r ❡✈❡r② ❛❧❣❡❜r❛ ❆ t❤❡ s❡t I ❧ ( ❆ ) ♦❢ ❛❧❧ ❧❡❢t ✐❞❡❛❧s ❛♥❞ t❤❡ s❡t I r ( ❆ ) ♦❢ ❛❧❧ r✐❣❤t ✐❞❡❛❧s✱ ♦r❞❡r❡❞ ❜② ✐♥❝❧✉s✐♦♥ ❛r❡ ❝♦♠♣❧❡t❡✱ ♠♦❞✉❧❛r ❧❛tt✐❝❡s ✇✐t❤ ♦♣❡r❛t✐♦♥s✿ ■ ∨ ❏ = ■ + ❏ ■ ∧ ❏ = ■ ∩ ❏ . ❛♥❞ ✭✶✮ ■♥ t❤❡s❡ ❧❛tt✐❝❡s ω = ✵ ❛♥❞ Ω = ❆ . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❆❣r❡❡♠❡♥ts ■❢ ❳ ⊆ ❆ ✐s ❛ s✉❜s❡t✱ t❤❡♥ ❧❡t ▲ ❆ ( ❳ ) = ▲ ( ❳ ) ❜❡ t❤❡ ❧❡❢t ❛♥♥✐❤✐❧❛t♦r ♦❢ ❳ ✐♥ ❆ ❛♥❞ ❧❡t ❘ ❆ ( ❳ ) = ❘ ( ❳ ) ❜❡ t❤❡ r✐❣❤t ❛♥♥✐❤✐❧❛t♦r ♦❢ ❳ ✐♥ ❆ : ▲ ( ❳ ) = { ❛ ∈ ❆ : ❛❳ = ✵ } , ✭✷✮ ❘ ( ❳ ) = { ❛ ∈ ❆ : ❳❛ = ✵ } . ✭✸✮ ❚❤❡♥ ▲ ( ❳ ) = ▲ ( ❘ ( ▲ ( ❳ ))) ❛♥❞ ❘ ( ❳ ) = ❘ ( ▲ ( ❘ ( ❳ ))) . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

▲❡t A ❧ ( ❆ ) ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ ❧❡❢t ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❆ ❛♥❞ A r ( ❆ ) ❜❡ t❤❡ s❡t ♦❢ ❛❧❧ r✐❣❤t ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❆ . ❚❤❡♥ A ❧ ( ❆ ) ⊆ I ❧ ( ❆ ) ✐s ❛ ❝♦♠♣❧❡t❡ ❧❛tt✐❝❡ ✇✐t❤ ♦♣❡r❛t✐♦♥s✿ ■ ∨ ❏ = ▲ ( ❘ ( ■ ) ∩ ❘ ( ❏ )) ■ ∧ ❏ = ■ ∩ ❏ , ❛♥❞ ❢♦r ■ , ❏ ∈ A ❧ ( ❆ ) , ω = ✵ ❛♥❞ Ω = ❆ . ❙✐♠✐❧❛r❧②✱ A r ( ❆ ) ⊆ I r ( ❆ ) ✐s ❛ ❝♦♠♣❧❡t❡ ❧❛tt✐❝❡ ✇✐t❤ ♦♣❡r❛t✐♦♥s ■ ∨ ❏ = ❘ ( ▲ ( ■ ) ∩ ▲ ( ❏ )) ■ ∧ ❏ = ■ ∩ ❏ , ❛♥❞ ❢♦r ■ , ❏ ∈ I r ( ❆ ) , ω = ✵ ❛♥❞ Ω = ❆ . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

▲❡t ❙ ❜❡ ❛ s❡♠✐❣r♦✉♣ ❛♥❞ ❧❡t ■ ❜❡ ❛♥ ✐❞❡❛❧ ✐♥ ❙ . ❚❤❡♥ t❤❡ ❘❡❡s ❢❛❝t♦r s❡♠✐❣r♦✉♣ ❙ / ■ ✐s ❡q✉❛❧ t♦ ❙ /ρ, ✇❤❡r❡ ρ ✐s t❤❡ ❝♦♥❣r✉❡♥❝❡ ♦♥ ❙ ❣✐✈❡♥ ❜② ( s , t ) ∈ ρ ✐❢ s = t ♦r s , t ∈ ■ . ✭✹✮ ▲❡t ❙ ❜❡ ❛ s❡♠✐❣r♦✉♣✳ ❚❤❡ s❡♠✐❣r♦✉♣ ❛❧❣❡❜r❛ K [ ❙ ] ✐s ❛ K ✲s♣❛❝❡ ✇✐t❤ t❤❡ ❜❛s✐s ❙ ❛♥❞ t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥❞✉❝❡❞ ❜② t❤❡ ♠✉❧t✐♣❧✐❝❛t✐♦♥ ✐♥ ❙ . ▲❡t ❙ ❜❡ s❡♠✐❣r♦✉♣ ✇✐t❤ ③❡r♦ ✵ . ❇② t❤❡ ❝♦♥tr❛❝t❡❞ s❡♠✐❣r♦✉♣ ❛❧❣❡❜r❛ ♦❢ ❙ ♦✈❡r K ✱ ❞❡♥♦t❡❞ ❜② K ✵ [ ❙ ] , ✇❡ ♠❡❛♥ t❤❡ ❢❛❝t♦r ❛❧❣❡❜r❛ K [ ❙ ] / K ✵ . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s

❇❛s✐❝ ❡①❛♠♣❧❡ ▲❡t P ❜❡ ❛ ✜♥✐t❡ ♣♦s❡t ❛♥❞ ❧❡t ▼ ( P ) ❜❡ t❤❡ ❢r❡❡ ♠♦♥♦✐❞ ✇✐t❤ t❤❡ s❡t P ♦❢ ❢r❡❡ ❣❡♥❡r❛t♦rs✳ ❈♦♥s✐❞❡r ✐♥ ▼ ( P ) ❛♥ ✐❞❡❛❧ ■ ❣❡♥❡r❛t❡❞ ❜② ❛❧❧ ♣r♦❞✉❝ts ①②③ ✇❤❡r❡ ① , ② , ③ ∈ P ❛♥❞ ❛❧❧ ♣r♦❞✉❝ts ①② ✇❤❡r❡ ① , ② ∈ P ❛♥❞ ① ≤ ② . P✉t P = ▼ ( P ) / ■ , t❤❡ ❘❡❡s ❢❛❝t♦r ♠♦♥♦✐❞✳ ■❢ P = ∅ t❤❡♥ ✇❡ ♣✉t P = ✶ . ■❢ P � = ∅ t❤❡♥ ❝❧❡❛r❧② P ⊆ P ✐♥ ❛ ♥❛t✉r❛❧ ✇❛② ❛♥❞ P ✷ = { ✵ } ∪ { ①② : ① , ② ∈ P , ① �≤ ② } . ▼♦r❡♦✈❡r✱ P = { ✶ } ∪ P ∪ P ✷ . ■❢ ① ✶ , ① ✷ , ② ✶ , ② ✷ ∈ P \ { ✶ } ❛r❡ s✉❝❤ t❤❛t ① ✶ ① ✷ = ② ✶ ② ✷ � = ✵ , t❤❡♥ ① ✶ = ② ✶ ❛♥❞ ① ✷ = ② ✷ . ▼❛➟❣♦r③❛t❛ ❏❛str③➛❜s❦❛ ❋✐♥✐t❡ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ✐♥ ❛❧❣❡❜r❛s ♦✈❡r ✜❡❧❞s