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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✐✈❡ P r♦❜❛❜✐❧✐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 ♥ ♥ ∈ N ℓ ♥ ◮ ❳ = �� � �� � ❝ ✵ ✭▲✕▲♦②✕❘❡❛❞ ✷✵✵✹✮ ❛♥❞ ✐ts ❞✉❛❧ ❳ = ✷ ✷ 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 ♥ ◮ �� � ❊ ❢♦r ✶ � q � ∞ ❛♥❞ ❊ = ❝ ✵ ♦r ❊ = ℓ ♣ ❢♦r ✶ � ♣ < ∞ q ✭▲✕❖❞❡❧❧✕❙❝❤❧✉♠♣r❡❝❤t✕❩sá❦ ✷✵✶✷❀ ▲❡✉♥❣ ✭ × ✷✮ ✷✵✶✹❀ ❑❛♥✐❛✕▲ ✷✵✶✹✮❀ ◮ �� � N ℓ q 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♣❛❝❡✬ ✭▼✐❝❤❛❧❛❦ ✷✵✵✸✮❀ ◮ ❈ [ ✵ , ω ω ] ❛♥❞ ❈ [ ✵ , α ] ✱ ✇❤❡r❡ α ✐s ❛ ❝♦✉♥t❛❜❧❡ ♦r❞✐♥❛❧ s❛t✐s❢②✐♥❣ α = ω α ✭❇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♣❛❝❡✳ ❚❤❡♥✿ ❈ ( ❑ ) ∼ = ℓ ♥ ◮ ❑ ❤❛s ♥ ∈ N ❡❧❡♠❡♥ts ⇐ ⇒ ∞ ❀ ❈ ( ❑ ) ∼ ◮ ✭▼✐❧✉t✐♥✮ ❑ ✐s ✉♥❝♦✉♥t❛❜❧❡ ⇐ ⇒ = ❈ [ ✵ , ✶ ] ❀ ◮ ✭❇❡ss❛❣❛ ❛♥❞ P❡➟❝③②➠s❦✐✮ ❑ ✐s ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ⇐ ⇒ ❈ ( ❑ ) ∼ = ❈ [ ✵ , ω ω α ] ❢♦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 ❈ [ ✵ , ω ω α ] ❢♦r ❝♦✉♥t❛❜❧❡ α ✳ ❚❤❡♦r❡♠ ✭❙❡♠❛❞❡♥✐ ✶✾✻✵✮✳ ❈ [ ✵ , ω ✶ ] ≇ ❈ [ ✵ , ω ✶ ] ⊕ ❈ [ ✵ , ω ✶ ] ✳ ✻