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■❞❡❛❧s ♦❢ ♦♣❡r❛t♦rs ♦♥ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ♦❢ ❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ t❤❡ ✜rst ✉♥❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧

◆✐❡❧s ▲❛✉sts❡♥

▲❛♥❝❛st❡r ❯♥✐✈❡rs✐t②✱ ❯❑

◗✉❛♥t✉♠ ●r♦✉♣s✱ ❖♣❡r❛t♦rs ❛♥❞ ◆♦♥✲❝♦♠♠✉t❛t✐✈❡ Pr♦❜❛❜✐❧✐t② ✷♥❞ s❡❛s♦♥✱ ✶st ♠❡❡t✐♥❣✿ ▲❛♥❝❛st❡r✱ ✷✺t❤ ❙❡♣t❡♠❜❡r ✷✵✶✹ ❏♦✐♥t ✇♦r❦ ✇✐t❤ ❚♦♠❛s③ ❑❛♥✐❛ ✭▲❛♥❝❛st❡r✮ ❛♥❞ P✐♦tr ❑♦s③♠✐❞❡r ✭■▼P❆◆✱ ❲❛rs❛✇✮

✶

▼♦t✐✈❛t✐♦♥✿ t❤❡ ✐❞❡❛❧ str✉❝t✉r❡ ♦❢ B(❳)

❋♦r ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳✱ ❝♦♥s✐❞❡r t❤❡ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ B(❳) = {❚ : ❳ → ❳ : ❚ ✐s ❜♦✉♥❞❡❞ ❛♥❞ ❧✐♥❡❛r}. ❖✈❡r❛❧❧ ❛✐♠✿ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❧❛tt✐❝❡ ♦❢ ✭❝❧♦s❡❞✱ t✇♦✲s✐❞❡❞✮ ✐❞❡❛❧s ♦❢ B(❳)✳ ❚❤✐s ✐s ❛ ✈❡r② ❞✐✣❝✉❧t ♣r♦❜❧❡♠❀ t❤❡ ♦♥❧② ❦♥♦✇♥ ❝♦♠♣❧❡t❡ ❝❧❛ss✐✜❝❛t✐♦♥s ❛r❡✿

◮ ❞✐♠ ❳ < ∞❀ ◮ ❳ = ℓ♣(I) ❢♦r ✶ ♣ < ∞ ❛♥❞ ❳ = ❝✵(I)✱ ✇❤❡r❡ I ✐s ❛♥ ❛r❜✐tr❛r② ✐♥✜♥✐t❡

✐♥❞❡① s❡t ✭❈❛❧❦✐♥ ✶✾✹✶❀ ●♦❤❜❡r❣✕▼❛r❦✉s✕❋❡❧❞♠❛♥ ✶✾✻✵❀ ●r❛♠s❝❤ ✶✾✻✼✴ ▲✉❢t ✶✾✻✽❀ ❉❛✇s ✷✵✵✻✮❀

◮ ❳ =

• ♥∈N ℓ♥

✷

• ❝✵ ✭▲✕▲♦②✕❘❡❛❞ ✷✵✵✹✮ ❛♥❞ ✐ts ❞✉❛❧ ❳ =
• ♥∈N ℓ♥

✷

• ℓ✶

✭▲✕❙❝❤❧✉♠♣r❡❝❤t✕❩sá❦ ✷✵✵✻✮❀

◮ ❆r❣②r♦s ❛♥❞ ❍❛②❞♦♥✬s ❇❛♥❛❝❤ s♣❛❝❡ ✇✐t❤ ✈❡r② ❢❡✇ ♦♣❡r❛t♦rs ✭✷✵✶✶✮✱

❛♥❞ s♦♠❡ ✈❛r✐❛♥ts ♦❢ ✐t ✭❚❛r❜❛r❞ ✷✵✶✷❀ ▼♦t❛❦✐s✕P✉❣❧✐s✐✕❩✐s✐♠♦♣♦✉❧♦✉ ✷✵✶✹❀ ❑❛♥✐❛✕▲ ✷✵✶✹✮❀

◮ ❳ = ❈(❑)✱ ✇❤❡r❡ ❑ ✐s t❤❡ ❵▼ró✇❦❛ s♣❛❝❡✬ ❝♦♥str✉❝t❡❞ ❜② ❑♦s③♠✐❞❡r

✭✷✵✵✺✮✱ ❛ss✉♠✐♥❣ ❈❍ ✭❇r♦♦❦❡r ✭✉♥♣✉❜❧✐s❤❡❞✮✴❑❛♥✐❛✕❑♦❝❤❛♥❡❦ ✷✵✶✹✮✳

✷

▼❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ B(❳)

❊❛s✐❡r ❣♦❛❧✿ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ B(❳) ❢♦r ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳✳ ◆♦t❡✿

◮ B(❳) ❛❧✇❛②s ❝♦♥t❛✐♥s ❛ ✉♥✐q✉❡ ♠✐♥✐♠❛❧ ♥♦♥✲③❡r♦ ✐❞❡❛❧✿ F(❳)✳ ◮ B(❳) ✐s ✉♥✐t❛❧✱ ❤❡♥❝❡✿

✕ ❡✈❡r② ♣r♦♣❡r ✐❞❡❛❧ ♦❢ B(❳) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧❀ ✕ t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧s ♦❢ B(❳) ❛r❡ ❛✉t♦♠❛t✐❝❛❧❧② ❝❧♦s❡❞✳ ❖❜s❡r✈❛t✐♦♥ ✭❉♦s❡✈✕❏♦❤♥s♦♥ ✷✵✶✵✮✳ ❚❤❡ s❡t M❳ = {❚ ∈ B(❳) : ∀ ❘, ❙ ∈ B(❳) : ■ = ❙❚❘} ✐s t❤❡ ✉♥✐q✉❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ B(❳) ✐❢ ✭❛♥❞ ♦♥❧② ✐❢✮ ✐t ✐s ❝❧♦s❡❞ ✉♥❞❡r ❛❞❞✐t✐♦♥✳

✸

❇❛♥❛❝❤ s♣❛❝❡s ❳ s✉❝❤ t❤❛t M❳ ✐s t❤❡ ✉♥✐q✉❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ B(❳)

❘❡❝❛❧❧✿ M❳ = {❚ ∈ B(❳) : ∀ ❘, ❙ ∈ B(❳) : ■ = ❙❚❘}✳

◮ ❝✵(I) ❛♥❞ ℓ♣(I) ❢♦r ✶ ♣ < ∞ ❛♥❞ ❛♥ ❛r❜✐tr❛r② ✐♥✜♥✐t❡ ✐♥❞❡① s❡t I✱

• ♥∈N ℓ♥

✷

• ❝✵ ❛♥❞
• ♥∈N ℓ♥

✷

• ℓ✶❀

◮ ♥∈N ℓ♥ q

• ❊ ❢♦r ✶ q ∞ ❛♥❞ ❊ = ❝✵ ♦r ❊ = ℓ♣ ❢♦r ✶ ♣ < ∞

✭▲✕❖❞❡❧❧✕❙❝❤❧✉♠♣r❡❝❤t✕❩sá❦ ✷✵✶✷❀ ▲❡✉♥❣ ✭×✷✮ ✷✵✶✹❀ ❑❛♥✐❛✕▲ ✷✵✶✹✮❀

◮ N ℓq

• ℓ♣ ❢♦r ✶ q < ♣ < ∞ ✭❈❤❡♥✕❏♦❤♥s♦♥✕❩❤❡♥❣ ✷✵✶✶✮❀

◮ ▲♦r❡♥t③ s❡q✉❡♥❝❡ s♣❛❝❡s ✭❑❛♠✐➠s❦❛✕P♦♣♦✈✕❙♣✐♥✉✕❚❝❛❝✐✉❝✕❚r♦✐ts❦②

✷✵✶✶✮❀

◮ ❝❡rt❛✐♥ ❖r❧✐❝③ s❡q✉❡♥❝❡ s♣❛❝❡s ✭▲✐♥✕❙❛r✐✕❩❤❡♥❣ ✷✵✶✹✮❀ ◮ t❤❡ q✉❛s✐✲r❡✢❡①✐✈❡ ❏❛♠❡s s♣❛❝❡s ❏♣ ❢♦r ✶ < ♣ < ∞ ✭▲ ✷✵✵✷✮❀ ◮ ❊❞❣❛r✬s ❧♦♥❣ ❏❛♠❡s s♣❛❝❡s ❏♣(ω✶) ❢♦r ✶ < ♣ < ∞ ✭❑❛♥✐❛✕❑♦❝❤❛♥❡❦ ✷✵✶✹✮❀ ◮ t❤❡ ❏❛♠❡s tr❡❡ s♣❛❝❡ ❛♥❞ t❤❡ ❏❛♠❡s ❢✉♥❝t✐♦♥ s♣❛❝❡ ✭❆♣❛ts✐❞✐s✕❆r❣②r♦s✕

❑❛♥❡❧❧♦♣♦✉❧♦s ✷✵✵✽✮✳

✹

❇❛♥❛❝❤ s♣❛❝❡s ❳ s✉❝❤ t❤❛t M❳ ✐s t❤❡ ✉♥✐q✉❡ ♠❛①✳ ✐❞❡❛❧ ♦❢ B(❳) ✭❝♦♥t✳✮

❘❡❝❛❧❧✿ M❳ = {❚ ∈ B(❳) : ∀ ❘, ❙ ∈ B(❳) : ■ = ❙❚❘}✳

◮ ▲♣[✵, ✶] ❢♦r ✶ ♣ < ∞ ✭❉♦s❡✈✕❏♦❤♥s♦♥✕❙❝❤❡❝❤t♠❛♥ ✷✵✶✶✮❀ ◮ ▲∞[✵, ✶] ∼

= ℓ∞ ✭▲✕▲♦② ✷✵✵✺✱ ✉s✐♥❣ P❡➟❝③②➠s❦✐ ❛♥❞ ❘♦s❡♥t❤❛❧✮❀

◮ ℓ∞(I) ❛♥❞ ℓ❝ ∞(I) ❢♦r ❛♥ ❛r❜✐tr❛r② ✐♥✜♥✐t❡ ✐♥❞❡① s❡t I

✭❏♦❤♥s♦♥✕❑❛♥✐❛✕❙❝❤❡❝❤t♠❛♥ ✷✵✶✹✮❀

◮ ℓ∞/❝✵ ✭✉s✐♥❣ ❉r❡✇♥♦✇s❦✐✕❘♦❜❡rts ✶✾✾✶✮❀ ◮ ❈[✵, ✶] ✭❇r♦♦❦❡r ✷✵✶✵✱ ✉s✐♥❣ P❡➟❝③②➠s❦✐ ❛♥❞ ❘♦s❡♥t❤❛❧✮❀ ◮ ❈(❆)✱ ✇❤❡r❡ ❆ ✐s t❤❡ ❵❞♦✉❜❧❡ ❛rr♦✇ s♣❛❝❡✬ ✭▼✐❝❤❛❧❛❦ ✷✵✵✸✮❀ ◮ ❈[✵, ωω] ❛♥❞ ❈[✵, α]✱ ✇❤❡r❡ α ✐s ❛ ❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧ s❛t✐s❢②✐♥❣ α = ωα

✭❇r♦♦❦❡r✱ ✉s✐♥❣ ❇♦✉r❣❛✐♥ ❛♥❞ P❡➟❝③②➠s❦✐✮❀

◮ ❈[✵, ω✶] ✭❑❛♥✐❛✕▲ ✷✵✶✷✮❀ ◮ α<ω✶ ❈[✵, α]

• ❝✵ ✭❑❛♥✐❛✕▲ ✷✵✶✹✮✳

✺

❈(❑)✲s♣❛❝❡s

❋♦r ❛ ❝♦♠♣❛❝t ❍❛✉s❞♦r✛ s♣❛❝❡ ❑✱ ❝♦♥s✐❞❡r t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❈(❑) = {❢ : ❑ → C : ❢ ✐s ❝♦♥t✐♥✉♦✉s}. ❋❛❝t✳ ❈(❑) s❡♣❛r❛❜❧❡ ⇐ ⇒ ❑ ♠❡tr✐③❛❜❧❡✳ ❈❧❛ss✐✜❝❛t✐♦♥✳ ▲❡t ❑ ❜❡ ❛ ❝♦♠♣❛❝t ♠❡tr✐❝ s♣❛❝❡✳ ❚❤❡♥✿

◮ ❑ ❤❛s ♥ ∈ N ❡❧❡♠❡♥ts

⇐ ⇒ ❈(❑) ∼ = ℓ♥

∞❀ ◮ ✭▼✐❧✉t✐♥✮

❑ ✐s ✉♥❝♦✉♥t❛❜❧❡ ⇐ ⇒ ❈(❑) ∼ = ❈[✵, ✶]❀

◮ ✭❇❡ss❛❣❛ ❛♥❞ P❡➟❝③②➠s❦✐✮

❑ ✐s ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ⇐ ⇒ ❈(❑) ∼ = ❈[✵, ωωα] ❢♦r ❛ ✉♥✐q✉❡ ❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧ α✳ ❍❡r❡✱ ❢♦r ❛♥ ♦r❞✐♥❛❧ σ✱ t❤❡ ✐♥t❡r✈❛❧ [✵, σ] = {α ♦r❞✐♥❛❧ : α σ} ✐s ❡q✉✐♣♣❡❞ ✇✐t❤ t❤❡ ♦r❞❡r t♦♣♦❧♦❣②✱ ✇❤✐❝❤ ✐s ❞❡t❡r♠✐♥❡❞ ❜② t❤❡ ❜❛s✐s [✵, β), (α, β), (α, σ] (✵ α < β σ). ◆♦t❡✿ ❈[✵, ω✶]✱ ✇❤❡r❡ ω✶ ✐s t❤❡ ✜rst ✉♥❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧✱ ✐s t❤❡ ✏♥❡①t✑ ❈(❑)✲s♣❛❝❡ ❛❢t❡r t❤❡ s❡♣❛r❛❜❧❡ ♦♥❡s ❈[✵, ωωα] ❢♦r ❝♦✉♥t❛❜❧❡ α✳ ❚❤❡♦r❡♠ ✭❙❡♠❛❞❡♥✐ ✶✾✻✵✮✳ ❈[✵, ω✶] ≇ ❈[✵, ω✶] ⊕ ❈[✵, ω✶]✳

✻

❚❤❡ t♦♣♦❧♦❣✐❝❛❧ ❞✐❝❤♦t♦♠②

❋♦r ❝♦♥✈❡♥✐❡♥❝❡✱ ❝♦♥s✐❞❡r t❤❡ ❤②♣❡r♣❧❛♥❡ ❈✵[✵, ω✶) = {❢ ∈ ❈[✵, ω✶] : ❢ (ω✶) = ✵} ✐♥st❡❛❞ ♦❢ ❈[✵, ω✶]✳ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕❑♦s③♠✐❞❡r✕▲✮✳ ▲❡t ❑ ❜❡ ❛ ✇❡❛❦∗✲❝♦♠♣❛❝t s✉❜s❡t ♦❢ ❈✵[✵, ω✶)∗✳ ❚❤❡♥ ❡①❛❝t❧② ♦♥❡ ♦❢ t❤❡ ❢♦❧❧♦✇✐♥❣ t✇♦ ❛❧t❡r♥❛t✐✈❡s ❤♦❧❞s✿

◮ ❑ ✐s ✉♥✐❢♦r♠❧② ❊❜❡r❧❡✐♥ ❝♦♠♣❛❝t✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❑ ✐s ❤♦♠❡♦♠♦r♣❤✐❝ t♦

❛ ✇❡❛❦❧② ❝♦♠♣❛❝t s✉❜s❡t ♦❢ ❛ ❍✐❧❜❡rt s♣❛❝❡❀

◮ ❑ ❝♦♥t❛✐♥s ❛ ❤♦♠❡♦♠♦r♣❤✐❝ ❝♦♣② ♦❢ [✵, ω✶] ♦❢ t❤❡ ❢♦r♠

{ρ + λδα : α ∈ ❉} ∪ {ρ}, ✇❤❡r❡ ρ ∈ ❈✵[✵, ω✶)∗✱ λ ∈ C \ {✵}✱ δα ✐s t❤❡ ❉✐r❛❝ ♠❡❛s✉r❡ ❛t α✱ ❛♥❞ ❉ ✐s ❛ ❝❧♦s❡❞ ❛♥❞ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ♦❢ [✵, ω✶)✳ ◆♦t❡✿ ✭✐✮ [✵, ω✶] ✐s ♥♦t ❝♦♥t❛✐♥❡❞ ✐♥ ❛♥② ✉♥✐❢♦r♠❧② ❊❜❡r❧❡✐♥ ❝♦♠♣❛❝t s♣❛❝❡❀ ✭✐✐✮ t❤❡ ✉♥✐t ❜❛❧❧ ♦❢ ❈✵[✵, ω✶)∗ ✐♥ t❤❡ ✇❡❛❦∗ t♦♣✳ ❝♦♥t❛✐♥s ❛ ❤♦♠❡♦♠♦r♣❤✐❝ ❝♦♣② ♦❢ ❡✈❡r② ✉♥✐❢♦r♠❧② ❊❜❡r❧❡✐♥ ❝♦♠♣❛❝t s♣❛❝❡ ♦❢ ❞❡♥s✐t② ❛t ♠♦st ℵ✶✳ ❖♣❡r❛t♦r✲t❤❡♦r❡t✐❝ ❛♣♣❧✐❝❛t✐♦♥✿ ❝♦♥s✐❞❡r ❑ = ❚ ∗(t❤❡ ✉♥✐t ❜❛❧❧ ♦❢ ❈✵[✵, ω✶)∗) ❢♦r ❚ ∈ B(❈✵[✵, ω✶))✳

✼

❈❤❛r❛❝t❡r✐③❛t✐♦♥s ♦❢ t❤❡ ✉♥✐q✉❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ B(❈✵[✵, ω✶))

❚❤❡♦r❡♠ ✭❑❛♥✐❛✕❑♦s③♠✐❞❡r✕▲✮✳ ▲❡t ❚ ∈ B(❈✵[✵, ω✶))✳ ❚❤❡♥ ❚❋❆❊✿ ✭❛✮ ❚ ∈ M❈✵[✵,ω✶) ✭t❤❛t ✐s✱ ■ = ❙❚❘ ❢♦r ❛❧❧ ❘, ❙ ∈ B(❈✵[✵, ω✶))✮❀ ✭❜✮ ❚ ❞♦❡s ♥♦t ✜① ❛ ❝♦♣② ♦❢ ❈✵[✵, ω✶); ✭❝✮ ❚ ✐s ❛ ❙❡♠❛❞❡♥✐ ♦♣❡r❛t♦r✱ ✐♥ t❤❡ s❡♥s❡ t❤❛t ❚ ∗∗ ♠❛♣s t❤❡ s✉❜s♣❛❝❡

• Λ ∈ ❈✵[✵, ω✶)∗∗ : λ♥, Λ → ✵ ❛s ♥ → ∞

❢♦r ❡✈❡r② ✇❡❛❦∗✲♥✉❧❧ s❡q✉❡♥❝❡ (λ♥) ✐♥ ❈✵[✵, ω✶)∗ ✐♥t♦ t❤❡ ❝❛♥♦♥✐❝❛❧ ❝♦♣② ♦❢ ❈✵[✵, ω✶) ✐♥ ✐ts ❜✐❞✉❛❧❀ ✭❞✮ t❤❡r❡ ✐s ❛ ❝❧♦s❡❞✱ ✉♥❜♦✉♥❞❡❞ s✉❜s❡t ❉ ♦❢ [✵, ω✶) s✉❝❤ t❤❛t (❚❢ )(α) = ✵ (❢ ∈ ❈✵[✵, ω✶), α ∈ ❉); ✭❡✮ ❚ ❢❛❝t♦rs t❤r♦✉❣❤ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡

• α<ω✶ ❈[✵, α]
• ❝✵;

✭❢✮ t❤❡ r❛♥❣❡ ♦❢ ❚ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❍✐❧❜❡rt✲❣❡♥❡r❛t❡❞ s✉❜s♣❛❝❡ ♦❢ ❈✵[✵, ω✶); t❤❛t ✐s✱ t❤❡r❡ ❡①✐st ❛ ❍✐❧❜❡rt s♣❛❝❡ ❍ ❛♥❞ ❛♥ ♦♣❡r❛t♦r ❯ : ❍ → ❈✵[✵, ω✶) s✉❝❤ t❤❛t ❚(❈✵[✵, ω✶)) ⊆ ❯(❍)❀ ✭❣✮ t❤❡ r❛♥❣❡ ♦❢ ❚ ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ✇❡❛❦❧② ❝♦♠♣❛❝t❧② ❣❡♥❡r❛t❡❞ s✉❜s♣❛❝❡ ♦❢ ❈✵[✵, ω✶); t❤❛t ✐s✱ t❤❡r❡ ❡①✐st ❛ r❡✢❡①✐✈❡ ❇❛♥❛❝❤ s♣❛❝❡ ❳ ❛♥❞ ❛♥ ♦♣❡r❛t♦r ❱ : ❳ → ❈✵[✵, ω✶) s✉❝❤ t❤❛t ❚(❈✵[✵, ω✶)) ⊆ ❱ (❳)✳

✽

❚❤❡ ❙③❧❡♥❦ ✐♥❞❡①

▲❡t ❳ ❜❡ ❛♥ ❆s♣❧✉♥❞ s♣❛❝❡ ✭t❤❛t ✐s✱ ❡✈❡r② s❡♣❛r❛❜❧❡ s✉❜s♣❛❝❡ ♦❢ ❳ ❤❛s s❡♣❛r❛❜❧❡ ❞✉❛❧✮✱ ❛♥❞ ❧❡t ❑ ⊂ ❳ ∗ ❜❡ ✇❡❛❦∗✲❝♦♠♣❛❝t✳ ❙③❧❡♥❦ ❛ss♦❝✐❛t❡❞ ❛♥ ♦r❞✐♥❛❧ ❙③ ❑ ✇✐t❤ ❑✱ ✐ts ❙③❧❡♥❦ ✐♥❞❡①✳ ❙❡t ❙③ ❳ = ❙③(t❤❡ ✉♥✐t ❜❛❧❧ ♦❢ ❳ ∗). ✭❲❡ ❡①t❡♥❞ t❤✐s t♦ ❛❧❧ ❇❛♥❛❝❤ s♣❛❝❡s ❜② ❙③ ❳ := ∞ ✇❤❡♥ ❳ ✐s ♥♦t ❆s♣❧✉♥❞✳✮ ❚❤❡♦r❡♠ ✭❙❛♠✉❡❧ ✶✾✽✸✮✳ ❙③ ❈[✵, ωωα] = ωα+✶ ❢♦r ❡❛❝❤ ❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧ α✳ ▼♦r❡ ❣❡♥❡r❛❧❧②✱ ❢♦r ❛♥ ♦♣❡r❛t♦r ❚ : ❳ → ❨ ✱ ❞❡✜♥❡ ❙③ ❚ = ❙③(❚ ∗(t❤❡ ✉♥✐t ❜❛❧❧ ♦❢ ❨ ∗)). ✭❚❤✐s ♠❛② ❛❧s♦ ❜❡ ∞✳✮ ❋♦r ❛♥ ♦r❞✐♥❛❧ α✱ s❡t SZ α(❳, ❨ ) = {❚ ∈ B(❳, ❨ ) : ❙③ ❚ ωα}. ❚❤❡♦r❡♠ ✭❇r♦♦❦❡r ✷✵✶✷✮✳ ❚❤❡ ❝❧❛ss SZ α ✐s ❛ ❝❧♦s❡❞✱ ✐♥❥❡❝t✐✈❡ ❛♥❞ s✉r❥❡❝t✐✈❡ ♦♣❡r❛t♦r ✐❞❡❛❧ ✐♥ t❤❡ s❡♥s❡ ♦❢ P✐❡ts❝❤ ❢♦r ❡✈❡r② ♦r❞✐♥❛❧ α✳

✾

❚❤❡ s❡❝♦♥❞✲❧❛r❣❡st ♣r♦♣❡r ✐❞❡❛❧ ♦❢ B(❈✵[✵, ω✶))

❙❡t ❊ω✶ =

• α<ω✶ ❈[✵, α]
• ❝✵✱ ❛♥❞ r❡❝❛❧❧ t❤❛t

❚ ∈ M❈✵[✵,ω✶) ⇐ ⇒ ❚ ❢❛❝t♦rs t❤r♦✉❣❤ ❊ω✶. ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕▲✮✳ ▲❡t ❚ ∈ B(❈✵[✵, ω✶))✳ ❚❤❡♥ ❚❋❆❊✿ ✭❛✮ ❚ ✜①❡s ❛ ❝♦♣② ♦❢ ❊ω✶❀ ✭❜✮ t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ♦♥ ❊ω✶ ❢❛❝t♦rs t❤r♦✉❣❤ ❚; ✭❝✮ t❤❡ ❙③❧❡♥❦ ✐♥❞❡① ♦❢ ❚ ✐s ✉♥❝♦✉♥t❛❜❧❡✳ ❈♦r♦❧❧❛r②✳ ❚❤❡ s❡t S❊ω✶ (❈✵[✵, ω✶)) =

• ❚ ∈ B(❈✵[✵, ω✶)) : ❚ ❞♦❡s ♥♦t ✜① ❛ ❝♦♣② ♦❢ ❊ω✶
• =
• ❚ ∈ B(❈✵[✵, ω✶)) : t❤❡ ✐❞❡♥t✐t② ♦♣❡r❛t♦r ♦♥ ❊ω✶

❞♦❡s ♥♦t ❢❛❝t♦r t❤r♦✉❣❤ ❚

• = {❚ ∈ B(❈✵[✵, ω✶)) : ❙③ ❚ < ω✶} =
• α<ω✶

SZ α(❈✵[✵, ω✶)) ✐s t❤❡ s❡❝♦♥❞✲❧❛r❣❡st ♣r♦♣❡r ❝❧♦s❡❞ ✐❞❡❛❧ ♦❢ B(❈✵[✵, ω✶))✿ ❢♦r ❡❛❝❤ ♣r♦♣❡r ✐❞❡❛❧ I ♦❢ B(❈✵[✵, ω✶))✱ ❡✐t❤❡r I = M❈✵[✵,ω✶) ♦r I ⊆ S❊ω✶ (❈✵[✵, ω✶))✳

✶✵

P❛rt✐❛❧ str✉❝t✉r❡ ♦❢ t❤❡ ❧❛tt✐❝❡ ♦❢ ❝❧♦s❡❞ ✐❞❡❛❧s ♦❢ B = B(❈✵[✵, ω✶))

X

X + G ❝✵(ω✶)

α<ω✶G ❝✵(ω✶, ❈(❑α))

S❊ω✶ M = G❊ω✶

• ✳

✳ ✳

• ✳

✳ ✳

• ✳

✳ ✳

• ✳

✳ ✳

• B

G ❈(❑α+✶)

• G ❈(❑α+✶)⊕❝✵(ω✶)
• G ❝✵(ω✶, ❈(❑α+✶))
• SZ α+✷
• G ❈(❑α)
• G ❈(❑α)⊕❝✵(ω✶)
• G ❝✵(ω✶, ❈(❑α))
• SZ α+✶
• ✳

✳ ✳

• ✳

✳ ✳

• ✳

✳ ✳

• ✳

✳ ✳

• G ❈(❑✶)
• G ❈(❑✶)⊕❝✵(ω✶)
• G ❝✵(ω✶, ❈(❑✶))
• SZ ✷
• G ❝✵
• G ❝✵(ω✶)
• SZ ✶
• K
• {✵}
• ❑α = [✵, ωωα], α < ω✶

✶✶

❈♦♥✈❡♥t✐♦♥s

◮ ❲❡ s✉♣♣r❡ss ❈✵[✵, ω✶) ❡✈❡r②✇❤❡r❡✱ t❤✉s ✇r✐t✐♥❣ K ✐♥st❡❛❞ ♦❢

K (❈✵[✵, ω✶)) ❢♦r t❤❡ ✐❞❡❛❧ ♦❢ ❝♦♠♣❛❝t ♦♣❡r❛t♦rs ♦♥ ❈✵[✵, ω✶)✱ ❡t❝✳❀

◮

I

J

♠❡❛♥s t❤❛t t❤❡ ✐❞❡❛❧ I ✐s ♣r♦♣❡r❧② ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ✐❞❡❛❧ J ❀

◮

I

J

✐♥❞✐❝❛t❡s t❤❛t t❤❡r❡ ❛r❡ ♥♦ ❝❧♦s❡❞ ✐❞❡❛❧s ❜❡t✇❡❡♥ I ❛♥❞ J ❀

◮ G❳ ❞❡♥♦t❡s t❤❡ s❡t ♦❢ ♦♣❡r❛t♦rs t❤❛t ❢❛❝t♦r t❤r♦✉❣❤ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❳

❛♥❞ G ❳ ✐ts ❝❧♦s✉r❡❀

◮ ❝✵(ω✶, ❳) ❞❡♥♦t❡s t❤❡ ❝✵✲❞✐r❡❝t s✉♠ ♦❢ ω✶ ❝♦♣✐❡s ♦❢ t❤❡ ❇❛♥❛❝❤ s♣❛❝❡ ❳✱

❛♥❞ ❝✵(ω✶) := ❝✵(ω✶, C)❀

◮ X ❞❡♥♦t❡s t❤❡ ✐❞❡❛❧ ♦❢ ♦♣❡r❛t♦rs ✇✐t❤ s❡♣❛r❛❜❧❡ r❛♥❣❡✳

✶✷

❇❛❝❦❣r♦✉♥❞✿ t❤❡ ❛✉t♦♠❛t✐❝ ❝♦♥t✐♥✉✐t② ♦❢ ❞❡r✐✈❛t✐♦♥s ❢r♦♠ B(❈[✵, ω✶])

❉❡✜♥✐t✐♦♥✳ ❆ ❧✐♥❡❛r ♠❛♣♣✐♥❣ δ ❢r♦♠ ❛ ❇❛♥❛❝❤ ❛❧❣❡❜r❛ A ✐♥t♦ ❛ ❇❛♥❛❝❤ A ✲❜✐♠♦❞✉❧❡ ✐s ❛ ❞❡r✐✈❛t✐♦♥ ✐❢ δ(❛❜) = ❛ · δ(❜) + δ(❛) · ❜ (❛, ❜ ∈ A ). ❚❤❡♦r❡♠ ✭❏♦❤♥s♦♥ ✶✾✻✼✮✳ ▲❡t ❳ ❜❡ ❛ ❇❛♥❛❝❤ s♣❛❝❡ s✉❝❤ t❤❛t ❳ ∼ = ❳ ⊕ ❳✳ ❚❤❡♥ ❡✈❡r② ❞❡r✐✈❛t✐♦♥ ❢r♦♠ B(❳) ✐♥t♦ ❛ ❇❛♥❛❝❤ B(❳)✲❜✐♠♦❞✉❧❡ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♥t✐♥✉♦✉s✳ ◗✉❡st✐♦♥✿ ✇❤❛t ❤❛♣♣❡♥s ✇❤❡♥ ❳ ≇ ❳ ⊕ ❳❄ ❚❤❡♦r❡♠ ✭▲♦②✕❲✐❧❧✐s ✶✾✽✾✮✳ ❊✈❡r② ❞❡r✐✈❛t✐♦♥ ❢r♦♠ B(❈[✵, ω✶]) ✐♥t♦ ❛ ❇❛♥❛❝❤ B(❈[✵, ω✶])✲❜✐♠♦❞✉❧❡ ✐s ❛✉t♦♠❛t✐❝❛❧❧② ❝♦♥t✐♥✉♦✉s✳ ❘❡♠❛r❦✳ ❆r♦✉♥❞ t❤❡ s❛♠❡ t✐♠❡✱ ❘❡❛❞ ❝♦♥str✉❝t❡❞ ❛ ❇❛♥❛❝❤ s♣❛❝❡ ❳ s✉❝❤ t❤❛t t❤❡r❡ ✐s ❛ ❞✐s❝♦♥t✐♥✉♦✉s ❞❡r✐✈❛t✐♦♥ ❢r♦♠ B(❳) ✐♥t♦ ❛ ❇❛♥❛❝❤ B(❳)✲❜✐♠♦❞✉❧❡✳

✶✸

❇♦✉♥❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐❞❡♥t✐t✐❡s ✐♥ t❤❡ ▲♦②✕❲✐❧❧✐s ✐❞❡❛❧

▲♦② ❛♥❞ ❲✐❧❧✐s✬ st❛rt✐♥❣ ♣♦✐♥t✿ B(❈[✵, ω✶]) ❝♦♥t❛✐♥s ❛ ♠❛①✐♠❛❧ ✐❞❡❛❧ M ♦❢ ❝♦❞✐♠❡♥s✐♦♥ ♦♥❡✳ ◆♦t❡✿ ♦✉r ✇♦r❦ s❤♦✇s t❤❛t M = M❈[✵,ω✶]✳ ❲❡ ❝❛❧❧ M t❤❡ ▲♦②✕❲✐❧❧✐s ✐❞❡❛❧✳ ▲♦② ❛♥❞ ❲✐❧❧✐s✬ ❦❡② st❡♣✿ M ❤❛s ❛ ❜♦✉♥❞❡❞ r✐❣❤t ❛♣♣r♦①✐♠❛t❡ ✐❞❡♥t✐t②✱ t❤❛t ✐s✱ ❛ ♥♦r♠✲❜♦✉♥❞❡❞ ♥❡t (❯❥) s✉❝❤ t❤❛t ❚❯❥ → ❚ ❢♦r ❡❛❝❤ ❚ ∈ M ✳ ◗✉❡st✐♦♥✿ ❞♦❡s M ❛❧s♦ ❤❛✈❡ ❛ ❜♦✉♥❞❡❞ ❧❡❢t ❛♣♣r♦①✐♠❛t❡ ✐❞❡♥t✐t②✱ t❤❛t ✐s✱ ❛ ♥♦r♠✲❜♦✉♥❞❡❞ ♥❡t (❯❥) s✉❝❤ t❤❛t ❯❥❚ → ❚ ❢♦r ❡❛❝❤ ❚ ∈ M ❄ ❆♥s✇❡r✿ ❨❡s✦ ✖ ■♥ ❢❛❝t ♠♦r❡ ✐s tr✉❡✿ ❚❤❡♦r❡♠ ✭❑❛♥✐❛✕❑♦s③♠✐❞❡r✕▲✮✳ M ❝♦♥t❛✐♥s ❛ ♥❡t (◗❥) ♦❢ ♣r♦❥❡❝t✐♦♥s ✇✐t❤ ◗❥ ✷ s✉❝❤ t❤❛t ∀ ❚ ∈ M ∃ ❥✵ ∀ ❥ ❥✵ : ◗❥❚ = ❚. ❈♦r♦❧❧❛r② ✭✉s✐♥❣ ❉✐①♦♥ ✶✾✼✸✮✳ M ❤❛s ❛ ❜♦✉♥❞❡❞ t✇♦✲s✐❞❡❞ ❛♣♣r♦①✐♠❛t❡ ✐❞❡♥t✐t②✳

✶✹

❆ ❢❡✇ r❡❢❡r❡♥❝❡s ✭✐♥ ❝❤r♦♥♦❧♦❣✐❝❛❧ ♦r❞❡r✮

◮ ❩✳ ❙❡♠❛❞❡♥✐✱ ❇❛♥❛❝❤ s♣❛❝❡s ♥♦♥✲✐s♦♠♦r♣❤✐❝ t♦ t❤❡✐r ❈❛rt❡s✐❛♥ sq✉❛r❡s✳ ■■✱

❇✉❧❧✳ ❆❝❛❞✳ P♦❧♦♥✳ ❙❝✐✳ ❙ér✳ ❙❝✐✳ ▼❛t❤✳ ❆str✳ P❤②s✳ ✽ ✭✶✾✻✵✮✱ ✽✶✕✽✹✳

◮ ❘✳ ❏✳ ▲♦② ❛♥❞ ●✳ ❆✳ ❲✐❧❧✐s✱ ❈♦♥t✐♥✉✐t② ♦❢ ❞❡r✐✈❛t✐♦♥s ♦♥ B(❊) ❢♦r ❝❡rt❛✐♥

❇❛♥❛❝❤ s♣❛❝❡s ❊✱ ❏✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✹✵ ✭✶✾✽✾✮✱ ✸✷✼✕✸✹✻✳

◮ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ❯♥✐q✉❡♥❡ss ♦❢ t❤❡ ♠❛①✐♠❛❧ ✐❞❡❛❧ ♦❢ t❤❡

❇❛♥❛❝❤ ❛❧❣❡❜r❛ ♦❢ ❜♦✉♥❞❡❞ ♦♣❡r❛t♦rs ♦♥ ❈([✵, ω✶])✱ ❏✳ ❋✉♥❝t✳ ❆♥❛❧✳ ✷✻✷ ✭✷✵✶✷✮✱ ✹✽✸✶✕✹✽✺✵❀ ❛r❳✐✈✿✶✶✶✷✳✹✽✵✵✳

◮ ❚✳ ❑❛♥✐❛✱ P✳ ❑♦s③♠✐❞❡r ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ❆ ✇❡❛❦∗✲t♦♣♦❧♦❣✐❝❛❧

❞✐❝❤♦t♦♠② ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ♦♣❡r❛t♦r t❤❡♦r②❀ ❚r❛♥s✳ ▲♦♥❞♦♥ ▼❛t❤✳ ❙♦❝✳ ✶✱ ✶✕✷✽ ✭✷✵✶✹✮❀ ❛r❳✐✈✿✶✸✵✸✳✵✵✷✵✳

◮ ❚✳ ❑❛♥✐❛ ❛♥❞ ◆✳ ❏✳ ▲❛✉sts❡♥✱ ❖♣❡r❛t♦rs ♦♥ t✇♦ ❇❛♥❛❝❤ s♣❛❝❡s ♦❢

❝♦♥t✐♥✉♦✉s ❢✉♥❝t✐♦♥s ♦♥ ❧♦❝❛❧❧② ❝♦♠♣❛❝t s♣❛❝❡s ♦❢ ♦r❞✐♥❛❧s❀ Pr♦❝✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ t♦ ❛♣♣❡❛r❀ ❛r❳✐✈✿✶✸✵✹✳✹✾✺✶✳

✶✺