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❈❧❛ss ❢♦r❝✐♥❣ ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝

❘❡❣✉❧❛ ❑r❛♣❢

❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥

❏❛♥✉❛r② ✶✷✱ ✷✵✶✼

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❘❡❣✉❧❛ ❑r❛♣❢

❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥

❏❛♥✉❛r② ✶✷✱ ✷✵✶✼

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷ ✴ ✶

❈♦♥t❡♥t

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✸ ✴ ✶

▼♦t✐✈❛t✐♦♥

❚❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s

❖♥❡ ♦❢ t❤❡ ♠♦st ♣r♦♠✐♥❡♥t ♦♣❡♥ ♣r♦❜❧❡♠s ✐♥ ♠❛t❤❡♠❛t✐❝s ✐♥ t❤❡ ❜❡❣✐♥♥✐♥❣ ♦❢ t❤❡ ✷✵t❤ ❝❡♥t✉r② ✇❛s t❤❡ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ✭❈❍✮ ❚❤❡r❡ ✐s ♥♦ s❡t ✇❤♦s❡ ❝❛r❞✐♥❛❧✐t② ✐s str✐❝t❧② ❜❡t✇❡❡♥ t❤❡ ❝❛r❞✐♥❛❧✐t② ♦❢ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs ❛♥❞ t❤❛t ♦❢ t❤❡ r❡❛❧ ♥✉♠❜❡rs✳ ■♥ ♦t❤❡r ✇♦r❞s✱ ✷ℵ✵ = ℵ✶✳ ❑✉rt ●ö❞❡❧ ♣r♦✈❡❞ ✐ts ❝♦♥s✐st❡♥❝② ✐♥ ✶✾✹✵ ❛♥❞ P❛✉❧ ❈♦❤❡♥ ♣r♦✈❡❞ ✐ts ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ❛①✐♦♠s ♦❢ ❩❋❈ ✐♥ ✶✾✻✸✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✹ ✴ ✶

▼♦t✐✈❛t✐♦♥

■♥❞❡♣❡♥❞❡♥❝❡ r❡s✉❧ts ✐♥ s❡t t❤❡♦r②

❉❡✜♥✐t✐♦♥ ❆ ♣r♦♣❡rt② Ψ ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ ❩❋❈✱ ✐❢ ♥❡✐t❤❡r Ψ ♥♦r ✐ts ♥❡❣❛t✐♦♥ ✐s ♣r♦✈❛❜❧❡ ❢r♦♠ ❩❋❈✳ ❚❤❡ ✇❛② t♦ ♣r♦✈❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ✐s t♦ ❝♦♥str✉❝t ♠♦❞❡❧s M, M′ ♦❢ ❩❋❈ s✉❝❤ t❤❛t M | = Ψ ❛♥❞ M′ | = ¬Ψ✳ ❆ ♠❡t❤♦❞ t♦ ❝♦♥str✉❝t s✉❝❤ ♠♦❞❡❧s ✐s ❢♦r❝✐♥❣✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✺ ✴ ✶

▼♦t✐✈❛t✐♦♥

❚❤❡ ✐❞❡❛ ♦❢ ❢♦r❝✐♥❣

❲❡ ❡①t❡♥❞ ❛ ❣✐✈❡♥ ♠♦❞❡❧ M ♦❢ ❩❋❈ t♦ ❛ ♥❡✇ ♠♦❞❡❧ M[G] ❜② ❛❞❞✐♥❣ ❛ ❣❡♥❡r✐❝ ♦❜❥❡❝t G✳ ❊✈❡r② ❡❧❡♠❡♥t ♦❢ M[G] ❤❛s ♥❛♠❡ σ ✐♥ M ✇❤✐❝❤ ✐s ❡✈❛❧✉❛t❡❞ ❛s σG✱ ✐✳❡✳ M[G] = {σG | σ ✐s ❛ ♥❛♠❡}. ❚❤✐s ✐s s✐♠✐❧❛r t♦ t❤❡ ❝❛s❡ ♦❢ ✜❡❧❞ ❡①t❡♥s✐♦♥s ✐♥ ❛❧❣❡❜r❛✿ ❈♦♥s✐❞❡r Q ❛♥❞ ❛♥ ❛❧❣❡❜r❛✐❝ ❝❧♦s✉r❡ ¯ Q ♦❢ Q✳ ■♥ Q✱ t❤❡ ♣♦❧②♥♦♠✐❛❧ X ✸ − ✷ ♥❛♠❡s t❤❡ r♦♦ts

✸

√ ✷, ζ

✸

√ ✷, ζ✷ ✸ √ ✷ ∈ ¯ Q ✭ζ = e

✷πi ✸ ✮✳

■❢ ✇❡ ❡①t❡♥❞ Q t♦ Q[

✸

√ ✷]✱ t❤❡♥

✸

√ ✷ ✐s t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ X ✸ − ✷✳ Q[

✸

√ ✷] ✐s ❛ ♠✐♥✐♠❛❧ ✜❡❧❞ ❡①t❡♥s✐♦♥ ♦❢ Q ✇✐t❤

✸

√ ✷ ∈ Q✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✻ ✴ ✶

▼♦t✐✈❛t✐♦♥

❋♦r❝✐♥❣

▲❡t M ❞❡♥♦t❡ ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ✭s♦♠❡ ❢r❛❣♠❡♥t ♦❢✮ ❩❋❈✳ ❋♦r❝✐♥❣ ✉s❡s ❛ ♣❛rt✐❛❧ ♦r❞❡r P = P, ≤P t♦ ❛♣♣r♦①✐♠❛t❡ t❤❡ ❣❡♥❡r✐❝ ♦❜❥❡❝t✳ ❊❧❡♠❡♥ts ♦❢ P ❛r❡ ❝❛❧❧❡❞ ❝♦♥❞✐t✐♦♥s✳ ❉❡✜♥✐t✐♦♥

✶ ❆ s✉❜s❡t D ⊆ P ✐s s❛✐❞ t♦ ❜❡ ❞❡♥s❡✱ ✐❢ ❢♦r ❡✈❡r② p ∈ P t❤❡r❡ ✐s s♦♠❡

q ≤P p ✇✐t❤ q ∈ D✳

✷ ❆ s✉❜s❡t G ⊆ P ✐s s❛✐❞ t♦ ❜❡ ❛ P✲❣❡♥❡r✐❝ ✜❧t❡r✱ ✐❢ ✐t ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣

♣r♦♣❡rt✐❡s✿

■❢ p ≤P q ❛♥❞ p ∈ G✱ t❤❡♥ q ∈ G✳ ■❢ p, q ∈ G t❤❡♥ t❤❡r❡ ✐s r ∈ G s✉❝❤ t❤❛t r ≤P p, q✳ ■❢ D ⊆ P ✐s ❛ ❞❡♥s❡ s❡t ✇❤✐❝❤ ✐s ✐♥ M✱ t❤❡♥ G ∩ D = ∅✳

❙✉❝❤ ✜❧t❡rs ❞♦ ♥♦t ❡①✐st ✐♥ M✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✼ ✴ ✶

▼♦t✐✈❛t✐♦♥

❈♦❤❡♥ ❢♦r❝✐♥❣

❇② ❛❞❞✐♥❣ ℵ✷✲♠❛♥② r❡❛❧s ✐♥st❡❛❞ ♦❢ ❥✉st ♦♥❡ ✇❡ ♦❜t❛✐♥ t❤❛t M[G] | = ¬❈❍✳ M M[G] M ❖r❞ ❖r❞ ℵ✷✲♠❛♥② ♥❡✇ r❡❛❧s ✷ℵ✵ = ℵ✷ ✷ℵ✵ = ℵ✶ ℵ✶ ℵ✷ σ ℵ✵ ℵ✷ σ ℵ✶ ℵ✵ G σG M[G] = {σG | σ✐s ❛ P✲♥❛♠❡}.

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✽ ✴ ✶

▼♦t✐✈❛t✐♦♥

❚❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠

▲❡t M ❜❡ ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ❩❋❈ ❛♥❞ P = P, ≤P ❛ ♣❛rt✐❛❧ ♦r❞❡r✳ ❚❤❡♦r❡♠ ■❢ G ✐s P✲❣❡♥❡r✐❝ ♦✈❡r M t❤❡♥ M[G] ✐s ❛ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ❩❋❈ ✇✐t❤ M ∪ {G} ⊆ M[G] ❛♥❞ ❖r❞M[G] = ❖r❞M✳ ▲❡t p ∈ P✱ ϕ(x) ❛ ❢♦r♠✉❧❛ ❛♥❞ σ ❛ P✲♥❛♠❡✳ ❲❡ s❛② t❤❛t p ❢♦r❝❡s ϕ(σ)✱ ❞❡♥♦t❡❞ p P ϕ(σ), ✐❢ ❢♦r ❡✈❡r② P✲❣❡♥❡r✐❝ ✜❧t❡r G ✇✐t❤ p ∈ G✱ M[G] | = ϕ(σG)✳ ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ r❡❧✐❡s ♦♥ ❚❤❡♦r❡♠ ✭❋♦r❝✐♥❣ t❤❡♦r❡♠✮

✶

❚❤❡ ❢♦r❝✐♥❣ r❡❧❛t✐♦♥ p M

P ϕ(σ) ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r M ✭❉❡✜♥❛❜✐❧✐t② ❧❡♠♠❛✮✳

✷

■❢ M[G] | = ϕ(σG) t❤❡♥ t❤❡r❡ ✐s p ∈ G s✉❝❤ t❤❛t p M

P ϕ(σ)

✭❚r✉t❤ ❧❡♠♠❛✮✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✾ ✴ ✶

▼♦t✐✈❛t✐♦♥

❋✉rt❤❡r ♣r♦♣❡rt✐❡s ♦❢ s❡t ❢♦r❝✐♥❣

✶ ❊✈❡r② ♣❛rt✐❛❧ ♦r❞❡r P ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✱ ✐✳❡✳ t❤❡r❡ ✐s ❛♥

✐♥❥❡❝t✐✈❡ ❞❡♥s❡ ❡♠❜❡❞❞✐♥❣ ❢r♦♠ P ✐♥t♦ s♦♠❡ ❝♦♠♣❧❡t❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛✳

✷ ■❢ t❤❡r❡ ✐s ❛ ❞❡♥s❡ ❡♠❜❡❞❞✐♥❣ P → Q✱ t❤❡♥ P✲❣❡♥❡r✐❝ ❡①t❡♥s✐♦♥s ❛♥❞

Q✲❣❡♥❡r✐❝ ❡①t❡♥s✐♦♥s ❝♦✐♥❝✐❞❡✳

✸ ❊✈❡r② s❡t ♦❢ ♦r❞✐♥❛❧s ✐♥ ❛ P✲❣❡♥❡r✐❝ ❡①t❡♥s✐♦♥ ❤❛s ❛ ♥✐❝❡ ♥❛♠❡✱ ✐✳❡✳ ❛

♥❛♠❡ ♦❢ t❤❡ ❢♦r♠

α<γ{ˇ

α} × Aα✱ ✇❤❡r❡ ❡❛❝❤ Aα ⊆ P ✐s ❛♥ ❛♥t✐❝❤❛✐♥✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✵ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❚❤❡ ❣❡♥❡r❛❧✐③❡❞ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ●❈❍

❈♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ str❡♥❣t❤❡♥✐♥❣ ♦❢ ❈❍✿

• ❡♥❡r❛❧✐③❡❞ ❈♦♥t✐♥✉✉♠ ❍②♣♦t❤❡s✐s ✭●❈❍✮

✷ℵα = ℵα+✶ ❢♦r ❡✈❡r② ♦r❞✐♥❛❧ α✳ ❚❤✐s ♠♦t✐✈❛t❡s t❤❡ ❢♦❧❧♦✇✐♥❣ ◗✉❡st✐♦♥ ■s ✐t ♣♦ss✐❜❧❡ t♦ ♦❜t❛✐♥ t❤❡ ●❈❍ ✭♦r ❢❛✐❧✉r❡s ❛t ❡✈❡r② ❧❡✈❡❧✮ ✉s✐♥❣ ❢♦r❝✐♥❣❄ ❯s✐♥❣ ❛ s❡t✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P ✐t ✐s ✐♠♣♦ss✐❜❧❡ t♦ ♠♦❞✐❢② t❤❡ ❝♦♥t✐♥✉✉♠ ❢✉♥❝t✐♦♥ ❛❜♦✈❡ ✷|P|✳ ❚❤❡ s♦❧✉t✐♦♥ ✐s t♦ ✉s❡ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡rs ✐♥st❡❛❞ ♦❢ s❡t✲s✐③❡❞ ♦♥❡s✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✶ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❆ ❣❡♥❡r❛❧ s❡tt✐♥❣ ❢♦r ❝❧❛ss ❢♦r❝✐♥❣

❲❡ st✉❞② ❝❧❛ss ❢♦r❝✐♥❣ ✐♥ ❛ s❡❝♦♥❞✲♦r❞❡r ❝♦♥t❡①t✳ ❉❡✜♥✐t✐♦♥ ▲❡t ❩❋− ❞❡♥♦t❡ ❩❋❈ ✇✐t❤♦✉t t❤❡ ♣♦✇❡r s❡t ❛①✐♦♠ ❛♥❞ t❤❡ ❛①✐♦♠ ♦❢ ❝❤♦✐❝❡✳ ❲❡ ❞❡♥♦t❡ ❜② ●❇− t❤❡ t❤❡♦r② ✐♥ t❤❡ t✇♦✲s♦rt❡❞ ❧❛♥❣✉❛❣❡ ✇✐t❤ ✈❛r✐❛❜❧❡s ❢♦r s❡ts ❛♥❞ ❝❧❛ss❡s✱ ✇✐t❤ s❡t ❛①✐♦♠s ❣✐✈❡♥ ❜② ❩❋− ✇✐t❤ ❝❧❛ss ♣❛r❛♠❡t❡rs ❛❧❧♦✇❡❞ ✐♥ t❤❡ s❝❤❡♠❛t❛ ♦❢ ❙❡♣❛r❛t✐♦♥ ❛♥❞ ❈♦❧❧❡❝t✐♦♥ ❝❧❛ss ❛①✐♦♠s ♦❢ ❡①t❡♥s✐♦♥❛❧✐t②✱ ❢♦✉♥❞❛t✐♦♥ ❛♥❞ ✜rst✲♦r❞❡r ❝❧❛ss ❝♦♠♣r❡❤❡♥s✐♦♥ ✭✐✳❡✳ ✐♥✈♦❧✈✐♥❣ ♦♥❧② s❡t q✉❛♥t✐✜❡rs✮✳ ❊①❛♠♣❧❡s ❛r❡ M, ❉❡❢(M)✱ ✇❤❡r❡ M ✐s ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ❩❋−✱ ❛♥❞ ♠♦❞❡❧s ♦❢ ❑❡❧❧❡②✲▼♦rs❡ ❝❧❛ss t❤❡♦r② ❑▼✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✷ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❈❧❛ss ❢♦r❝✐♥❣ ❡①t❡♥s✐♦♥s

▲❡t M = M, C ❜❡ ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ●❇−✳ ❲❡ ✇♦r❦ ✇✐t❤ ♣❛rt✐❛❧ ♦r❞❡rs P = P, ≤P s✉❝❤ t❤❛t ≤P, P ∈ C✳ MP ❞❡♥♦t❡s t❤❡ s❡t ♦❢ P✲♥❛♠❡s ✇❤✐❝❤ ❛r❡ ✐♥ M ✭s❡t ♥❛♠❡s✮✳ CP ❞❡♥♦t❡s t❤❡ s❡t ♦❢ P✲♥❛♠❡s ✇❤✐❝❤ ❛r❡ ✐♥ C ✭❝❧❛ss ♥❛♠❡s✮✳ ❆ ✜❧t❡r G ✐s P✲❣❡♥❡r✐❝ ♦✈❡r M ✐❢ ✐t ♠❡❡ts ❛❧❧ ❞❡♥s❡ s✉❜❝❧❛ss❡s ♦❢ M ✇❤✐❝❤ ❛r❡ ✐♥ C✳ ❊✈❛❧✉❛t✐♦♥s ♦❢ ♥❛♠❡s ❛r❡ ❞❡✜♥❡❞ ❛s ✉s✉❛❧✳ ❲❡ s❡t M[G] = M[G], C[G]✱ ✇❤❡r❡ M[G] = {σG | σ ∈ MP} C[G] = {ΓG | Γ ∈ CP}✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✸ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❚❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠

▲❡t P ❜❡ ❛ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r✱ σ ∈ MP ❛♥❞ Γ ∈ CP✳ ❲❡ ✇r✐t❡ p M

P ϕ(σ, Γ) ✐❢ ❢♦r ❡✈❡r② P✲❣❡♥❡r✐❝ ✜❧t❡r G ✇✐t❤ p ∈ G✱

M[G] | = ϕ(σG, ΓG)✳ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t P s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ♦✈❡r M✱ ✐❢ ❢♦r ❡✈❡r② L∈✲❢♦r♠✉❧❛ ϕ(x, C) ❛❧❧♦✇✐♥❣ ❝❧❛ss ♣❛r❛♠❡t❡rs ❛♥❞ ❢♦r ❡✈❡r② Γ ∈ CP✱

✶ {p, σ ∈ P × MP | p M

P ϕ(σ, Γ)} ∈ C ✭❞❡✜♥❛❜✐❧✐t② ❧❡♠♠❛✮

✷ ✇❤❡♥❡✈❡r G ✐s P✲❣❡♥❡r✐❝ ♦✈❡r M✱ σ ∈ MP ❛♥❞ Γ ∈ CP s✉❝❤ t❤❛t

M[G] | = ϕ(σG, ΓG) t❤❡♥ t❤❡r❡ ✐s p ∈ G ✇✐t❤ p M

P ϕ(σ, Γ)

✭tr✉t❤ ❧❡♠♠❛✮✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✹ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❆ ❢❛✐❧✉r❡ ♦❢ ❘❡♣❧❛❝❡♠❡♥t

■♥ t❤❡ ❢♦❧❧♦✇✐♥❣✱ ✇❡ ♣r❡s❡♥t s♦♠❡ ❡①❛♠♣❧❡s ✇❤✐❝❤ ✐❧❧✉str❛t❡ t❤❛t ❝❧❛ss ❢♦r❝✐♥❣ ❜❡❤❛✈❡s ❞✐✛❡r❡♥t❧② ❢r♦♠ s❡t ❢♦r❝✐♥❣✳ ❋r♦♠ ♥♦✇ ♦♥✱ ❧❡t M = M, C ❞❡♥♦t❡ ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ●❇−✳ ❖❜s❡r✈❛t✐♦♥ ▲❡t P = ❈♦❧(ω, ❖r❞) ❞❡♥♦t❡ t❤❡ ♣❛rt✐❛❧ ♦r❞❡r ✇❤♦s❡ ❝♦♥❞✐t✐♦♥s ❛r❡ ✜♥✐t❡ ❢✉♥❝t✐♦♥s p : ❞♦♠(p) → ❖r❞M✱ ❞♦♠(p) ⊆ ω ✜♥✐t❡✱ ♦r❞❡r❡❞ ❜② r❡✈❡rs❡ ✐♥❝❧✉s✐♦♥✳ ❚❤❡♥ P ❛❞❞s ❛ s✉r❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ω → ❖r❞M✳ ■♥ ♣❛rt✐❝✉❧❛r✱ ❘❡♣❧❛❝❡♠❡♥t ❢❛✐❧s ✐♥ t❤❡ ❣❡♥❡r✐❝ ❡①t❡♥s✐♦♥✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✺ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❋❛✐❧✉r❡s ♦❢ t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠

✳✳✳ ❜✉t ✐t ❝❛♥ ❣❡t ❡✈❡♥ ✇♦rs❡✿ ❚❤❡♦r❡♠ ✭❍♦❧②✱ ❑✳✱ ▲ü❝❦❡✱ ◆❥❡❣♦♠✐r✱ ❙❝❤❧✐❝❤t ✷✵✶✺✮ ▲❡t M ❜❡ ❛ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧ ♦❢ ❩❋−✳ ❚❤❡r❡ ✐s ❛ ♣❛rt✐❛❧ ♦r❞❡r P ⊆ M ✇❤✐❝❤ ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r M s✉❝❤ t❤❛t P ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ ❞❡✜♥❛❜✐❧✐t② ❧❡♠♠❛ ♦✈❡r M✳ ✳✳✳ ❛♥❞ ❡✈❡♥ ✇♦rs❡ t❤❛♥ t❤❛t✿ ❚❤❡♦r❡♠ ✭❍♦❧②✱ ❑✳✱ ▲ü❝❦❡✱ ◆❥❡❣♦♠✐r✱ ❙❝❤❧✐❝❤t ✷✵✶✺✮ ❚❤❡r❡ ❛r❡ ❝♦✉♥t❛❜❧❡ tr❛♥s✐t✐✈❡ ♠♦❞❡❧s M ♦❢ ❩❋− ❢♦r ✇❤✐❝❤ t❤❡r❡ ✐s ❛ ♣❛rt✐❛❧ ♦r❞❡r P t❤❛t ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r M s✉❝❤ t❤❛t P ❞♦❡s ♥♦t s❛t✐s❢② t❤❡ tr✉t❤ ❧❡♠♠❛ ♦✈❡r M✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✻ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥s

❉❡✜♥✐t✐♦♥

✶ ❆ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ B ✐s M✲❝♦♠♣❧❡t❡ ✐❢ t❤❡ s✉♣r❡♠✉♠ s✉♣B A ♦❢ ❛❧❧

❡❧❡♠❡♥ts ✐♥ A ❡①✐sts ✐♥ B ❢♦r ❡✈❡r② A ∈ M ✇✐t❤ A ⊆ B✳

✷ ❲❡ s❛② t❤❛t P ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥ ✐♥ M ✐❢ t❤❡r❡ ✐s ❛♥

M✲❝♦♠♣❧❡t❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ B s✉❝❤ t❤❛t ✐ts ❞♦♠❛✐♥✱ ❛❧❧ ❇♦♦❧❡❛♥ ♦♣❡r❛t✐♦♥s ♦❢ B ❛r❡ ✐♥ C ❛♥❞ t❤❡r❡ ✐s ❛♥ ✐♥❥❡❝t✐✈❡ ❞❡♥s❡ ❡♠❜❡❞❞✐♥❣ ❢r♦♠ P ✐♥t♦ B \ {✵B} ✐♥ C✳ ■♥ s❡t ❢♦r❝✐♥❣✱ ❡✈❡r② ♣❛rt✐❛❧ ♦r❞❡r ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✳ ❚❤❡♦r❡♠ ✭❍♦❧②✱ ❑✳✱ ▲ü❝❦❡✱ ◆❥❡❣♦♠✐r✱ ❙❝❤❧✐❝❤t✮ ❙✉♣♣♦s❡ t❤❛t C ❝♦♥t❛✐♥s ❛ ❣❧♦❜❛❧ ✇❡❧❧✲♦r❞❡r✳ ❚❤❡♥ ❛ ♣❛rt✐❛❧ ♦r❞❡r P s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡r❡ ❛r❡ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡rs ✇❤✐❝❤ ❞♦ ♥♦t ❤❛✈❡ ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✼ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

❋❛✐❧✉r❡s ♦❢ t❤❡ ❡①t❡♥s✐♦♥ ♠❛①✐♠❛❧✐t② ♣r✐♥❝✐♣❧❡

❉❡✜♥✐t✐♦♥ ❆ ♣❛rt✐❛❧ ♦r❞❡r P s❛t✐s✜❡s t❤❡ ❡①t❡♥s✐♦♥ ♠❛①✐♠❛❧✐t② ♣r✐♥❝✐♣❧❡ ✭❊▼P✮ ♦✈❡r M | = ●❇− ✐❢ ❢♦r ❡✈❡r② ♣❛rt✐❛❧ ♦r❞❡r Q s✉❝❤ t❤❛t P ✐s ❞❡♥s❡ ✐♥ Q ❛♥❞ ❢♦r ❡✈❡r② Q✲❣❡♥❡r✐❝ ✜❧t❡r G ♦✈❡r M✱ M[G] = M[G ∩ P]✳ ❊✈❡r② s❡t✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r s❛t✐s✜❡s t❤❡ ❊▼P✳ ❖❜s❡r✈❛t✐♦♥ ▲❡t ❈♦❧∗(ω, ❖r❞) ❞❡♥♦t❡ t❤❡ s✉❜♦r❞❡r ♦❢ ❈♦❧(ω, ❖r❞) ♦❢ ❝♦♥❞✐t✐♦♥s p ✇✐t❤ ❞♦♠(p) ∈ ω✳ ❈❧❡❛r❧②✱ ❈♦❧∗(ω, ❖r❞) ✐s ❞❡♥s❡ ✐♥ ❈♦❧(ω, ❖r❞)✳ ❍♦✇❡✈❡r✱ ❈♦❧(ω, ❖r❞) ❝♦❧❧❛♣s❡s ❛❧❧ M✲❝❛r❞✐♥❛❧s ❜✉t ❈♦❧∗(ω, ❖r❞) ❞♦❡s ♥♦t ❛❞❞ ❛♥② ♥❡✇ s❡ts✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✽ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

◆♦♥✲❡①✐st❡♥❝❡ ♦❢ ♥✐❝❡ ♥❛♠❡s

▲❡t P ❜❡ ❛ ♣❛rt✐❛❧ ♦r❞❡r✳ ❆ ♥✐❝❡ ♥❛♠❡ ✐s ❛ P✲♥❛♠❡ ♦❢ t❤❡ ❢♦r♠

• α<γ{ˇ

α} × Aα✱ ✇❤❡r❡ Aα ⊆ P ✐s ❛♥ ❛♥t✐❝❤❛✐♥ ✐♥ M ❛♥❞ γ ∈ ❖r❞M✳ ❉❡✜♥✐t✐♦♥ P ✐s ♥✐❝❡✱ ✐❢ ❢♦r ❡✈❡r② γ ∈ ❖r❞M, σ ∈ MP ❛♥❞ ❢♦r ❡✈❡r② P✲❣❡♥❡r✐❝ ✜❧t❡r G s✉❝❤ t❤❛t σG ⊆ γ t❤❡r❡ ✐s ❛ ♥✐❝❡ ♥❛♠❡ τ ∈ MP ✇✐t❤ σG = τ G✳ ❈♦♥s✐❞❡r t❤❡ ❢♦r❝✐♥❣ ♥♦t✐♦♥ P = ❈♦❧(ω, ❖r❞) ❛♥❞ σ = {ˇ n, {n, ✵} | n ∈ ω}✳ ❚❤❡ s❡t ω \ σG ❤❛s ❛ ♥❛♠❡✱ ❜✉t ♥♦ ♥✐❝❡ ♥❛♠❡✿ ❙✉♣♣♦s❡ t❤❛t µ =

n∈ω{ˇ

n} × An ❛♥❞ p P µ = ˇ ω \ σ✳ ❚❛❦❡ n / ∈ ❞♦♠(p) ❛♥❞ α > r❛♥❦(An) ❛♥❞ ♣✉t q = p ∪ {n, α}✳ ❚❤❡♥ q P ˇ n ∈ µ s♦ t❤❡r❡ ♠✉st ❜❡ r ∈ An ✇❤✐❝❤ ✐s ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ q✳ ❇✉t t❤❡♥ n ∈ ❞♦♠(r) ❛♥❞ s♦ r(n) = α✱ ❛ ❝♦♥tr❛❞✐❝t✐♦♥✳ ❈♦♥❝❧✉s✐♦♥ ❈♦❧(ω, ❖r❞) ✐s ♥♦t ♥✐❝❡✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✾ ✴ ✶

❈❧❛ss ❢♦r❝✐♥❣

▼♦t✐✈❛t✐♦♥

❲❡ ♥❡❡❞ t♦ ♣❧❛❝❡ s♦♠❡ r❡str✐❝t✐♦♥s ♦♥ t❤❡ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡rs ✉s❡❞ ❢♦r ❢♦r❝✐♥❣✳ ◗✉❡st✐♦♥

✶ ❯♥❞❡r ✇❤❛t ❝♦♥❞✐t✐♦♥s ❞♦❡s ❛ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r s❛t✐s❢② t❤❡

❢♦r❝✐♥❣ t❤❡♦r❡♠ ❛♥❞ ♣r❡s❡r✈❡ t❤❡ ❛①✐♦♠s ♦❢ ●❇−❄

✷ ■s t❤❡r❡ ❛ ✭♠✐♥✐♠❛❧✮ ♣r♦♣❡rt② ✇❤✐❝❤ ❡♥s✉r❡s t❤❛t ❛❧❧ ♣r♦♣❡rt✐❡s ♦❢ s❡t

❢♦r❝✐♥❣ ❝❛♥ ❜❡ tr❛♥s❢❡rr❡❞ t♦ ❝❧❛ss ❢♦r❝✐♥❣❄

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✵ ✴ ✶

Pr❡t❛♠❡♥❡ss

Pr❡t❛♠❡♥❡ss

❚❤❡ ❢♦❧❧♦✇✐♥❣ ♥♦t✐♦♥ ✇❛s ✐♥tr♦❞✉❝❡❞ ❜② ❙② ❋r✐❡❞♠❛♥✳ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P = P, ≤P ✐s ♣r❡t❛♠❡ ❢♦r M = M, C ✐❢ ❢♦r ❡✈❡r② p ∈ P ❛♥❞ ❢♦r ❡✈❡r② s❡q✉❡♥❝❡ ♦❢ ❞❡♥s❡ ❝❧❛ss❡s Di | i ∈ I ∈ C ✇✐t❤ I ∈ M t❤❡r❡ ✐s q ≤P p ❛♥❞ di | i ∈ I ∈ M s✉❝❤ t❤❛t ❢♦r ❡✈❡r② i ∈ I✱ di ⊆ Di ❛♥❞ di ✐s ♣r❡❞❡♥s❡ ❜❡❧♦✇ q✳ Pr❡t❛♠❡♥❡ss ❛❧❧♦✇s ✉s t♦ ♣❛ss ❢r♦♠ ❞❡♥s❡ ❝❧❛ss❡s t♦ ♣r❡❞❡♥s❡ s❡ts ❜② str❡♥❣t❤❡♥✐♥❣ ❛ ❣✐✈❡♥ ❝♦♥❞✐t✐♦♥✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✶ ✴ ✶

Pr❡t❛♠❡♥❡ss

Pr❡t❛♠❡♥❡ss

❚❤❡♦r❡♠ ✭❙✳ ❋r✐❡❞♠❛♥✮ ▲❡t M ❜❡ ❛ ♠♦❞❡❧ ♦❢ ●❇− s✉❝❤ t❤❛t ❡✐t❤❡r M | = t❤❡ ♣♦✇❡r s❡t ❛①✐♦♠✱ ♦r C ❝♦♥t❛✐♥s ❛ s❡t✲❧✐❦❡ ✇❡❧❧✲♦r❞❡r✳ ❚❤❡♥ t❤❡ ❢♦❧❧♦✇✐♥❣ st❛t❡♠❡♥ts ❤♦❧❞ ❢♦r ❡✈❡r② ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P✿

✶ ■❢ P ✐s ♣r❡t❛♠❡ t❤❡♥ P s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳ ✷ ■❢ P ✐s ♣r❡t❛♠❡ ✐✛ M[G] |

= ●❇− ❢♦r ❡✈❡r② P✲❣❡♥❡r✐❝ ✜❧t❡r G✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✷ ✴ ✶

Pr❡t❛♠❡♥❡ss

Pr❡t❛♠❡♥❡ss ❛♥❞ t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠

◆♦t❛t✐♦♥ ▲❡t Ψ ❜❡ s♦♠❡ ♣r♦♣❡rt② ♦❢ ♣❛rt✐❛❧ ♦r❞❡rs✳ ❲❡ s❛② t❤❛t ❛ ♣❛rt✐❛❧ ♦r❞❡r P s❛t✐s✜❡s Ψ ❞❡♥s❡❧②✱ ✐❢ ❡✈❡r② ♣❛rt✐❛❧ ♦r❞❡r Q s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ ❞❡♥s❡ ❡♠❜❡❞❞✐♥❣ ❢r♦♠ P ✐♥t♦ Q s❛t✐s✜❡s Ψ✳ ❲❡ ❤❛✈❡ s❡❡♥ t❤❛t t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ♠❛② ❢❛✐❧ ❢♦r ❝❧❛ss ❢♦r❝✐♥❣✳ ❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ t❤❡r❡ ❛r❡ ♥♦♥✲♣r❡t❛♠❡ ♣❛rt✐❛❧ ♦r❞❡rs s✉❝❤ ❛s ❈♦❧(ω, ❖r❞) ✇❤✐❝❤ ❞♦ s❛t✐s❢② t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳ ❚❤❡♦r❡♠ ✭❍♦❧②✱ ❑✳✱ ❙❝❤❧✐❝❤t✮ ❙✉♣♣♦s❡ t❤❛t M | = ●❇− ❛♥❞ C ❝♦♥t❛✐♥s ❛ s❡t✲❧✐❦❡ ✇❡❧❧✲♦r❞❡r ❜✉t ♥♦ ✜rst✲♦r❞❡r tr✉t❤ ♣r❡❞✐❝❛t❡✳ ❚❤❡♥ ❛ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P ✐s ♣r❡t❛♠❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐t ❞❡♥s❡❧② s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✸ ✴ ✶

Pr❡t❛♠❡♥❡ss

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠

Pr❡t❛♠❡♥❡ss ♦❢ ❛ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P ✐s ✲ ✉♥❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ M ✲ ❡q✉✐✈❛❧❡♥t t♦ ❡❛❝❤ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ ♣r♦♣❡rt✐❡s✿ P ♣r❡s❡r✈❡s t❤❡ ❛①✐♦♠s ♦❢ ●❇−✳ P ♣r❡s❡r✈❡s ❙❡♣❛r❛t✐♦♥✴❘❡♣❧❛❝❡♠❡♥t✴❈♦❧❧❡❝t✐♦♥✳ P ❞♦❡s ♥♦t ❛❞❞ ❛ ❝♦✜♥❛❧✴s✉r❥❡❝t✐✈❡ ❢✉♥❝t✐♦♥ ❢r♦♠ s♦♠❡ ♦r❞✐♥❛❧ κ ✐♥t♦ ❖r❞M✳ P s❛t✐s✜❡s t❤❡ ❊▼P✳ P ❞❡♥s❡❧② s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳ P ✐s ❞❡♥s❡❧② ♥✐❝❡✳ P ❞❡♥s❡❧② ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✳ ❆❧❧ ♣r♦♣❡rt✐❡s ❛❜♦✈❡ ❛❧✇❛②s ❤♦❧❞ ❢♦r s❡t✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡rs❀ t❤✐s s✉❣❣❡sts t❤❛t ♣r❡t❛♠❡ ❢♦r❝✐♥❣s ❛r❡ t❤❡ ✏r✐❣❤t✑ ❝❧❛ss ♦❢ ❝❧❛ss ❢♦r❝✐♥❣s t♦ ❝♦♥s✐❞❡r✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✹ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ t❤❡ ❖r❞✲❝❝

❆ ❝❧❛ss✲s✐③❡❞ ♣❛rt✐❛❧ ♦r❞❡r P ✐s s❛✐❞ t♦ s❛t✐s❢② t❤❡ ❖r❞✲❝❝✱ ✐❢ ❛❧❧ ✐ts ❛♥t✐❝❤❛✐♥s ❛r❡ ❡❧❡♠❡♥ts ♦❢ M✳ ❲❡ ❝❛♥ str❡♥❣❤t❡♥ ♠❛♥② ♣r❡✈✐♦✉s❧② ❝♦♥s✐❞❡r❡❞ ♣r♦♣❡rt✐❡s ❛♥❞ ♦❜t❛✐♥ ❝❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ t❤❡ ❖r❞✲❝❝✳ ❚❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ✲ ✉♥❞❡r s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s ♦♥ t❤❡ ❣r♦✉♥❞ ♠♦❞❡❧ M ✲ ❡q✉✐✈❛❧❡♥t t♦ P s❛t✐s❢②✐♥❣ t❤❡ ❖r❞✲❝❝✿ P s❛t✐s✜❡s t❤❡ str♦♥❣ ❡①t❡♥s✐♦♥ ♠❛①✐♠❛❧✐t② ♣r✐♥❝✐♣❧❡✳ P s❛t✐s✜❡s t❤❡ ♠❛①✐♠❛❧✐t② ♣r✐♥❝✐♣❧❡✳ P ✐s ❞❡♥s❡❧② ✈❡r② ♥✐❝❡✳ P ❤❛s ❛ ✉♥✐q✉❡ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥✳ P ❤❛s ❛ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥ B s✉❝❤ t❤❛t ❡✈❡r② s✉❜❝❧❛ss ♦❢ B ✇❤✐❝❤ ✐s ✐♥ C ❤❛s ❛ s✉♣r❡♠✉♠ ✐♥ B✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✺ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

❖t❤❡r ♣r♦♣❡rt✐❡s ✇❤✐❝❤ ✐♠♣❧② t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠

❲❡ ❤❛✈❡ ❢♦✉♥❞ ♦t❤❡r ♣r♦♣❡rt✐❡s t❤❛♥ ♣r❡t❛♠❡♥❡ss ✇❤✐❝❤ ✐♠♣❧② t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳ ❉❡✜♥✐t✐♦♥ ❲❡ s❛② t❤❛t ❛ ♣❛rt✐❛❧ ♦r❞❡r P = P, ≤P ❤❛s t❤❡ s❡t ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✱ ✐❢ ❢♦r ❡✈❡r② p ∈ P ❛♥❞ ❡✈❡r② s❡t A ⊆ P ✐♥ M✱ t❤❡r❡ ✐s ❛♥ ❡①t❡♥s✐♦♥ q ≤P p ♦❢ p s✉❝❤ t❤❛t ❢♦r ❡✈❡r② a ∈ A✱ ❡✐t❤❡r q ≤P a ♦r q⊥Pa✳ ❚❤❡♦r❡♠ ✭❍♦❧②✱ ❑✳✱ ❙❝❤❧✐❝❤t✮

✶ P ❤❛s t❤❡ s❡t ❞❡❝✐s✐♦♥ ♣r♦♣❡rt② ✐✛ ✐t ❞♦❡s♥✬t ❛❞❞ ❛♥② ♥❡✇ s❡ts✳ ✷ ■❢ P ❤❛s t❤❡ s❡t ❞❡❝✐s✐♦♥ ♣r♦♣❡rt②✱ t❤❡♥ ✐t s❛t✐s✜❡s t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠✳ ❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✻ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

P❡t❡r ❍♦❧②✱ ❘❡❣✉❧❛ ❑r❛♣❢✱ P❤✐❧✐♣♣ ▲ü❝❦❡✱ ❆♥❛ ◆❥❡❣♦♠✐r ❛♥❞ P❤✐❧✐♣♣ ❙❝❤❧✐❝❤t✳ ❈❧❛ss ❢♦r❝✐♥❣✱ t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ❛♥❞ ❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥s✳ ❏♦✉r♥❛❧ ♦❢ ❙②♠❜♦❧✐❝ ▲♦❣✐❝ ✽✶✱ ♥♦✳ ✹✱ ♣♣✳ ✶✺✵✵✲✶✺✸✵✱ ✷✵✶✻✳ P❡t❡r ❍♦❧②✱ ❘❡❣✉❧❛ ❑r❛♣❢ ❛♥❞ P❤✐❧✐♣♣ ❙❝❤❧✐❝❤t✳ ❙❡♣❛r❛t✐♦♥ ❛♥❞ ❘❡♣❧❛❝❡♠❡♥t ✐♥ ❝❧❛ss ❢♦r❝✐♥❣ ❡①t❡♥s✐♦♥s✳ ❙✉❜♠✐tt❡❞✳ P❡t❡r ❍♦❧②✱ ❘❡❣✉❧❛ ❑r❛♣❢ ❛♥❞ P❤✐❧✐♣♣ ❙❝❤❧✐❝❤t✳ ❊q✉✐✈❛❧❡♥❝❡s ♦❢ ♣r❡t❛♠❡♥❡ss ❛♥❞ t❤❡ ❖r❞✲❝❝✳ ❆❝❝❡♣t❡❞ ❢♦r t❤❡ ❆♥♥❛❧s ♦❢ P✉r❡ ❛♥❞ ❆♣♣❧✐❡❞ ▲♦❣✐❝✱ ✷✵✶✻✳ ❘❡❣✉❧❛ ❑r❛♣❢✳ ❈❧❛ss ❢♦r❝✐♥❣ ❛♥❞ s❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝✳ P❤❉ t❤❡s✐s✱ ✷✵✶✻✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✼ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

❙❡❝♦♥❞✲♦r❞❡r ❛r✐t❤♠❡t✐❝ ❛♥❞ ❩❋❈

❚❤❡♦r❡♠ ❙❖❆ ❛♥❞ ❩❋❈− + ❱ = ❍❈ ❜✐✲✐♥t❡r♣r❡t ❡❛❝❤ ♦t❤❡r✳ ❇② ✉s✐♥❣ t❤❡ Π✶

✶✲P❙P t♦ ♣r♦✈❡ t❤❛t ▲ |

= ❩❋❈✱ ❛♥❞ ✉s✐♥❣ ❛ ❝❧❛ss ✈❡rs✐♦♥ ♦❢ t❤❡ ▲é✈② ❝♦❧❧❛♣s❡ ♦♥❡ ❝❛♥ ♣r♦✈❡ ❚❤❡♦r❡♠ ✭❑♦❡♣❦❡✲▼♦❡❧❧❡r❢❡❧❞✮ ❙❖❆ + Π✶

✶✲P❙P ❛♥❞ ❩❋❈ ❛r❡ ❡q✉✐❝♦♥s✐st❡♥t✳

▲❡t ❩❋❈# ❞❡♥♦t❡ t❤❡ t❤❡♦r② ❩❋❈+ ✏❡✈❡r② s❡t ♦❢ ♦r❞✐♥❛❧s ❤❛s ❛ s❤❛r♣✑✳ ❚❤❡♦r❡♠ ✭❑✳✮ ❙❖❆ + Π✶

✶✲❉❡t + Π✶ ✷✲P❙P ❛♥❞ ❩❋❈# ❛r❡ ❡q✉✐❝♦♥s✐st❡♥t✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✽ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

❈❧❛ss ❢♦r❝✐♥❣ ♦✈❡r ❙❖❆

Pr❡t❛♠❡ ❝❧❛ss ❢♦r❝✐♥❣ ♦✈❡r ❙❖❆ ❝♦rr❡s♣♦♥❞s t♦ ❝❧❛ss ❢♦r❝✐♥❣ ✇✐t❤ ♣r❡t❛♠❡ ♣❛rt✐❛❧ ♦r❞❡rs ✇❤✐❝❤ s❛t✐s❢② t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ♦✈❡r ♠♦❞❡❧s ♦❢ ❩❋❈−✳ ❆❧❧ ✏❝♦♠♠♦♥✑ tr❡❡ ❢♦r❝✐♥❣s ❤❛✈❡ ❛ ♣r❡✲❇♦♦❧❡❛♥ ❝♦♠♣❧❡t✐♦♥ s❛t✐s❢② t❤❡ ❢♦r❝✐♥❣ t❤❡♦r❡♠ ❛r❡ ♣r❡t❛♠❡✳ ❚❤✐s ❛❞❞✐t✐♦♥❛❧❧② ✉s❡s ❞❡♣❡♥❞❡♥t ❝❤♦✐❝❡ ❉❈✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ tr❡❡ ❢♦r❝✐♥❣s ♣r❡s❡r✈❡ ❙❖❆✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✷✾ ✴ ✶

❋✉rt❤❡r r❡s✉❧ts

Pr❡s❡r✈❛t✐♦♥ ♦❢ t❤❡ ♣❡r❢❡❝t s❡t ♣r♦♣❡rt②

❚❤❡♦r❡♠ ✭❈❛st✐❜❧❛♥❝♦✲❙❝❤❧✐❝❤t✮ ❆❧❧ ✏❝♦♠♠♦♥✑ tr❡❡ ❢♦r❝✐♥❣s s✉❝❤ ❛s ❙❛❝❦s ❢♦r❝✐♥❣ ♦r ▼❛t❤✐❛s ❢♦r❝✐♥❣ ♣r❡s❡r✈❡ t❤❡ Π✶

✶✲P❙P ♦✈❡r ♠♦❞❡❧s ♦❢ ❙❖❆ + ❉❈✳

❖♥ t❤❡ ♦t❤❡r ❤❛♥❞✱ ✉s✐♥❣ ❛❧♠♦st ❞✐s❥♦✐♥t ❝♦❞✐♥❣ ❛♥❞ r❡s❤❛♣✐♥❣✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣✿ ❚❤❡♦r❡♠ ✭❑✳✮ ❚❤❡r❡ ✐s ❛ ❢♦r❝✐♥❣ ♥♦t✐♦♥ P s✉❝❤ t❤❛t ✐♥ ❡✈❡r② P✲❣❡♥❡r✐❝ ❡①t❡♥s✐♦♥ M[G] t❤❡r❡ ✐s ❛ r❡❛❧ x ✇✐t❤ M[G] = L[x]✳ ■♥ ♣❛rt✐❝✉❧❛r✱ t❤❡ Π✶

✶[x]✲P❙P ❢❛✐❧s ✐♥

M[G]✳

❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✸✵ ✴ ✶