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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 σ G ✱ ✐✳❡✳ M [ G ] = { σ G | 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 ✷ π i ✸ ✮✳ ✭ ζ = e √ √ ✷ ✐s t❤❡ ❡✈❛❧✉❛t✐♦♥ ♦❢ X ✸ − ✷✳ ✸ ✸ ■❢ ✇❡ ❡①t❡♥❞ Q t♦ Q [ ✷ ] ✱ t❤❡♥ √ √ ✸ ✸ 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 G ❖r❞ ❖r❞ σ σ σ G ℵ ✷ ℵ ✷ ℵ ✶ ℵ ✶ ℵ ✵ ℵ ✵ ℵ ✷ ✲♠❛♥② ♥❡✇ r❡❛❧s ✷ ℵ ✵ = ℵ ✶ ✷ ℵ ✵ = ℵ ✷ M [ G ] = { σ G | 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 ) , = ϕ ( σ G ) ✳ ✐❢ ❢♦r ❡✈❡r② P ✲❣❡♥❡r✐❝ ✜❧t❡r G ✇✐t❤ p ∈ G ✱ M [ G ] | ❚❤❡ ♣r♦♦❢ ♦❢ t❤❡ t❤❡♦r❡♠ ❛❜♦✈❡ r❡❧✐❡s ♦♥ ❚❤❡♦r❡♠ ✭❋♦r❝✐♥❣ t❤❡♦r❡♠✮ ❚❤❡ ❢♦r❝✐♥❣ r❡❧❛t✐♦♥ p � M P ϕ ( σ ) ✐s ❞❡✜♥❛❜❧❡ ♦✈❡r M ✭❉❡✜♥❛❜✐❧✐t② ❧❡♠♠❛✮ ✳ ✶ = ϕ ( σ G ) t❤❡♥ t❤❡r❡ ✐s p ∈ G s✉❝❤ t❤❛t p � M ■❢ M [ G ] | 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 ❛ ♥✐❝❡ ♥❛♠❡✱ ✐✳❡✳ ❛ α<γ { ˇ α } × A α ✱ ✇❤❡r❡ ❡❛❝❤ A α ⊆ P ✐s ❛♥ ❛♥t✐❝❤❛✐♥✳ ♥❛♠❡ ♦❢ t❤❡ ❢♦r♠ � ❘❡❣✉❧❛ ❑r❛♣❢ ✭❯♥✐✈❡rs✐t② ♦❢ ❇♦♥♥✮ ❈❧❛ss ❢♦r❝✐♥❣ ❏❛♥✉❛r② ✶✷✱ ✷✵✶✼ ✶✵ ✴ ✶