t tts trs - - PowerPoint PPT Presentation

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

❇❛s✐❝ ❡①❛♠♣❧❡ ◆♦✇ ❧❡t K(P) = K✵[P] ❜❡ t❤❡ ❝♦♥tr❛❝t❡❞ ♠♦♥♦✐❞ ❛❧❣❡❜r❛✳ ❚❤✉s P ⊂ K(P) ❛♥❞ K(P) ❤❛s t❤❡ ♥❛t✉r❛❧ ❣r❛❞❛t✐♦♥ ❣✐✈❡♥ ❜②✿ K(P) = K ⊕ ❱ ⊕ ❱ ✷, ✭✺✮ ✇❤❡r❡ t❤❡ ♥❛t✉r❛❧ ❜❛s❡ ♦❢ ❱ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ P ❛♥❞ t❤❡ ♥❛t✉r❛❧ ❜❛s❡ ♦❢ ❱ ✷ ❝❛♥ ❜❡ ✐❞❡♥t✐✜❡❞ ✇✐t❤ P✷ \ {✵}. ❖✉r ❛❧❣❡❜r❛ K(P) ✐s ❛ ❧♦❝❛❧✱ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✱ ✇✐t❤ t❤❡ ❏❛❝♦❜s♦♥ r❛❞✐❝❛❧ ❏ = ❱ ⊕ ❱ ✷ ❛♥❞ ✇✐t❤ t❤❡ r❡s✐❞✉❡ ✜❡❧❞ K(P)/❏ = K.

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

■❢ ▲ ✐s ❛ ❧❛tt✐❝❡ t❤❡♥ ✇❡ ♣✉t K▲ = K(P), ✇❤❡r❡ P = ▲ \ {ω, Ω}. ❚❤❡♦r❡♠ ▲❡t P ❜❡ ❛♥② ✜♥✐t❡ ♣♦s❡t ❛♥❞ ❧❡t φ : P − → A❧(K(P)) ❜❡ ❣✐✈❡♥ ❜② φ(①) = ▲K(P)(①) ❢♦r ① ∈ P. ❚❤❡♥ φ ✐s ❛♥ ❡♠❜❡❞❞✐♥❣ ❛♥❞ ♣r❡s❡r✈❡s ❛❧❧ ❡①✐st✐♥❣ ♠❡❡ts ❛♥❞ ❥♦✐♥s✳ ■❢ ▲ ✐s ❛ ✜♥✐t❡ ❧❛tt✐❝❡✱ t❤❡♥ φ ❡①t❡♥❞s ✉♥✐q✉❡❧② t♦ ❛ ❧❛tt✐❝❡ ✐s♦♠♦r♣❤✐s♠ ♦❢ ▲ ✇✐t❤ t❤❡ ✐♥t❡r✈❛❧ [φ(ω), φ(Ω)] ⊆ A❧(K▲).

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

❈♦r♦❧❧❛r② ▲❡t ▲ ❜❡ ❛ ✜♥✐t❡ ❧❛tt✐❝❡ s✉❝❤ t❤❛t t❤❡ s❡t P = ▲ \ {Ω, ω} ❤❛s t❤❡ s♠❛❧❧❡st ❛♥❞ t❤❡ ❧❛r❣❡st ❡❧❡♠❡♥t✳ ❚❤❡♥ φ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ ❧❛tt✐❝❡s ▲ ❛♥❞ A❧(K▲)✳ ❈♦r♦❧❧❛r② ❚❤❡r❡ ✐s ♥♦ ♥♦♥tr✐✈✐❛❧ ❧❛tt✐❝❡ ✐❞❡♥t✐t② s❛t✐s✜❡❞ ✐♥ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ♦❢ ❛❧❧ ✜♥✐t❡ ❞✐♠❛♥s✐♦♥❛❧ ❛❧❣❡❜r❛s✳

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

❈♦r♦❧❧❛r② ▲❡t ▲ ❜❡ ❛ ✜♥✐t❡ ❧❛tt✐❝❡ s✉❝❤ t❤❛t t❤❡ s❡t P = ▲ \ {Ω, ω} ❤❛s t❤❡ s♠❛❧❧❡st ❛♥❞ t❤❡ ❧❛r❣❡st ❡❧❡♠❡♥t✳ ❚❤❡♥ φ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ♦❢ ❧❛tt✐❝❡s ▲ ❛♥❞ A❧(K▲)✳ ❈♦r♦❧❧❛r② ❚❤❡r❡ ✐s ♥♦ ♥♦♥tr✐✈✐❛❧ ❧❛tt✐❝❡ ✐❞❡♥t✐t② s❛t✐s✜❡❞ ✐♥ ❧❛tt✐❝❡s ♦❢ ❛♥♥✐❤✐❧❛t♦rs ♦❢ ❛❧❧ ✜♥✐t❡ ❞✐♠❛♥s✐♦♥❛❧ ❛❧❣❡❜r❛s✳

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

❚❤❡♦r❡♠ ▲❡t P ❜❡ ❛ ♣♦s❡t ✇✐t❤ |P| = ♠ < ∞. ❯♥❞❡r t❤❡ ♥♦t❛t✐♦♥ ❢r♦♠ t❤❡ ❛❜♦✈❡ t❤❡♦r❡♠ ✇❡ ❤❛✈❡ ✶ + ♠(♠ + ✶) ✷ ≤ ❉✐♠K(K(P)) ≤ ✶ + ♠✷. ■❢ ▲ ✐s ❛ ❧❛tt✐❝❡ ✇✐t❤ |▲| = ♥ < ∞, t❤❡♥ ✶ + (♥ − ✷)(♥ − ✶) ✷ ≤ ❉✐♠K(K▲) ≤ ✶ + (♥ − ✷)✷.

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

▲❡t ▲ ❜❡ t❤❡ ❧❛tt✐❝❡ ω

s ◗ ◗

• s

s s ✑✑s ❅ ❅ ❅

Ω ❚❤❡♥ K▲ = K{❛, ❜, ❝}/(❛✷, ❛❝, ❝✷, ❜✷, ❉) ✇❤❡r❡ ❉ = {①②③|①, ②, ③ ∈ {❛, ❜, ❝}}. ❚❤✉s ❉✐♠K(K▲) = ✾. ❍♦✇❡✈❡r✱ t❤❡ ❧❛tt✐❝❡ ▲ ❝❛♥ ❜❡ ❛❧s♦ ❡♠❜❡❞❞❡❞ ✐♥t♦ t❤❡ ❧❛tt✐❝❡ A❧(❆)✱ ✇❤❡r❡ ❆ = K{①, ②, ③}/(①✷, ①②, ② ✷, ③✷, ①③ − ③①, ②③ − ③②, ❉)✱ ✇❤❡r❡ ❉ ✐s ❛s ❛❜♦✈❡✱ ❜✉t ❉✐♠K(❆) = ✼.

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

❚❤❡♦r❡♠ ▲❡t ❆ ❜❡ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛❧❣❡❜r❛ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ K ✇✐t❤ A❧(❆) ✜♥✐t❡✳ ❚❤❡♥ ❆ ✐s ❛ ✜♥✐t❡ ❞✐r❡❝t s✉♠ ♦❢ ❧♦❝❛❧ ❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ▲❡t ▲ ❜❡ ❛ ✜♥✐t❡ ❧❛tt✐❝❡✳ ❚❤❡♥ ▲

❧ ❆ ❢♦r ❛♥

❛❧❣❡❜r❛ ❆ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ▲ ✐s ❛ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❧❛tt✐❝❡s ✇✐t❤ ✉♥✐q✉❡ ❛t♦♠s ❛♥❞ ❝♦❛t♦♠s✳

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

❚❤❡♦r❡♠ ▲❡t ❆ ❜❡ ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧ ❛❧❣❡❜r❛ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ K ✇✐t❤ A❧(❆) ✜♥✐t❡✳ ❚❤❡♥ ❆ ✐s ❛ ✜♥✐t❡ ❞✐r❡❝t s✉♠ ♦❢ ❧♦❝❛❧ ❛❧❣❡❜r❛s✳ ❚❤❡♦r❡♠ ▲❡t ▲ ❜❡ ❛ ✜♥✐t❡ ❧❛tt✐❝❡✳ ❚❤❡♥ ▲ ≃ A❧(❆) ❢♦r ❛♥ ❛❧❣❡❜r❛ ❆ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ K ✐❢ ❛♥❞ ♦♥❧② ✐❢ ▲ ✐s ❛ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ✐♥❞❡❝♦♠♣♦s❛❜❧❡ ❧❛tt✐❝❡s ✇✐t❤ ✉♥✐q✉❡ ❛t♦♠s ❛♥❞ ❝♦❛t♦♠s✳

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

❈♦r♦❧❧❛r② ▲❡t ♥ ≥ ✶✳ ▲❡t ❇✷♥ ❞❡♥♦t❡ t❤❡ ❜♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ ❝❛r❞✐♥❛❧✐t② ✷♥. ❚❤❡♥ ❇✷♥ ≃ A❧(❆) ❢♦r ❛♥ ❛❧❣❡❜r❛ ❆ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ K ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❆ ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ ♥ ❞✐✈✐s✐♦♥ ❛❧❣❡❜r❛s ♦✈❡r K✳ ❊①❛♠♣❧❡ ▲❡t r ♥ ✶ ❈♦♥s✐❞❡r ❆ ①✶ ①r ■ ♥ ✇❤❡r❡ ■ ✐s t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② ①✶ ①r ❚❤❡♥ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t ❆ ✐s ❛ ❧♦❝❛❧ ❛❧❣❡❜r❛ ✇✐t❤

❧ ❆ t❤❡ ❝❤❛✐♥

♦❢ ♥ ❡❧❡♠❡♥ts✳ ❋♦r ❡✈❡r② r ♥ ✶ ♦✉r ❛❧❣❡❜r❛ ❆ ❤❛s t❤❡ ❞✐♠❡♥s✐♦♥ r ♥

✶ r ✶

r ✇❤✐❝❤ ❝❛♥ ❜❡ q✉✐t❡ ❛r❜✐tr❛r②✳

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

❈♦r♦❧❧❛r② ▲❡t ♥ ≥ ✶✳ ▲❡t ❇✷♥ ❞❡♥♦t❡ t❤❡ ❜♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ ❝❛r❞✐♥❛❧✐t② ✷♥. ❚❤❡♥ ❇✷♥ ≃ A❧(❆) ❢♦r ❛♥ ❛❧❣❡❜r❛ ❆ ♦✈❡r ❛♥ ✐♥✜♥✐t❡ ✜❡❧❞ K ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❆ ✐s t❤❡ ❞✐r❡❝t s✉♠ ♦❢ ♥ ❞✐✈✐s✐♦♥ ❛❧❣❡❜r❛s ♦✈❡r K✳ ❊①❛♠♣❧❡ ▲❡t r, ♥ ≥ ✶. ❈♦♥s✐❞❡r ❆ = K{①✶, . . . , ①r}/■ ♥, ✇❤❡r❡ ■ ✐s t❤❡ ✐❞❡❛❧ ❣❡♥❡r❛t❡❞ ❜② ①✶, . . . ①r. ❚❤❡♥ ✐t ✐s ❡❛s② t♦ ❝❤❡❝❦ t❤❛t ❆ ✐s ❛ ❧♦❝❛❧ ❛❧❣❡❜r❛ ✇✐t❤ A❧(❆) t❤❡ ❝❤❛✐♥ ♦❢ ♥ ❡❧❡♠❡♥ts✳ ❋♦r ❡✈❡r② r, ♥ > ✶ ♦✉r ❛❧❣❡❜r❛ ❆ ❤❛s t❤❡ ❞✐♠❡♥s✐♦♥ r ♥−✶

r−✶ > r, ✇❤✐❝❤ ❝❛♥ ❜❡ q✉✐t❡ ❛r❜✐tr❛r②✳

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

❚❤❡♦r❡♠ ❋♦r ❡✈❡r② ✜♥✐t❡ ♣♦s❡t P t❤❡r❡ ❡①✐sts ❛ ✜♥✐t❡ ❞✐♠❡♥s✐♦♥❛❧✱ ❝♦♠♠✉t❛t✐✈❡ ❛❧❣❡❜r❛ AP ❛♥❞ ❛♥ ❡♠♠❜❡❞❞✐♥❣ ♦❢ P ✐♥t♦ A❧(AP) ♣r❡s❡r✈✐♥❣ ❛❧❧ ❡①✐st✐♥❣ ♠❡❡ts ❛♥❞ ❥♦✐♥s✳ ■❢ ▲ ✐s ❛ ✜♥✐t❡ ❧❛tt✐❝❡ t❤❡♥ ▲ ❝❛♥ ❜❡ ❝♦♥s✐❞❡r❡❞ ❛s ❛ s✉❜❧❛tt✐❝❡ ♦❢ A❧(AL).

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

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

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