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❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ▼❱✲s❡♠✐♠♦❞✉❧❡s ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦ P❤❉s ✐♥ ▲♦❣✐❝ ❳■✱ ❯♥✐✈❡rs✐t② ♦❢ ❇❡r♥ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✶ ✴ ✷✶

▼❱✲❆❧❣❡❜r❛s ❖♥ ❆r♦✉♥❞ ✶✾✺✽ ❈✳ ❈✳ ❈❤❛♥❣ ❞❡✈✐s❡❞ ▼❱✲❛❧❣❡❜r❛s t♦ st✉❞② ■♥❥❡❝t✐✈✐t② ❛♥❞ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ ✐♥ ♣❛rt✐❝✉❧❛r ▼❱✲❛❧❣❡❜r❛s ❢♦r♠ t❤❡ ❛❧❣❡❜r❛✐❝ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠❛♥t✐❝s ♦❢ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❉❡✜♥✐t✐♦♥ ✭▼❱✲❛❧❣❡❜r❛✮ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ❆♥ ▼❱✲❛❧❣❡❜r❛ ✐s ❛♥ ❛❧❣❡❜r❛ ( A , ⊕ , ∗ , ✵ ) ♦❢ t②♣❡ ✭✷✱✶✱✵✮ s✉❝❤ ✉♥✐s❛✳✐t t❤❛t✱ ❢♦r ❡✈❡r② x , y , z ∈ A ✇❡ ❤❛✈❡✿ ✶ ( x ⊕ y ) ⊕ z = x ⊕ ( y ⊕ z ) ❀ ✷ x ⊕ y = y ⊕ x ❀ ✸ x ⊕ ✵ = x ❀ ✹ ( x ∗ ) ∗ = x ❀ ✺ x ⊕ ✵ ∗ = ✵ ∗ ❀ ✻ ( x ∗ ⊕ y ) ∗ ⊕ y = ( y ∗ ⊕ x ) ∗ ⊕ x ✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✷ ✴ ✷✶

❊①❛♠♣❧❡s ♦❢ ▼❱✲❛❧❣❡❜r❛s ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ ❊①❛♠♣❧❡ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ ✶ ❚❤❡ r❡❛❧ ✐♥t❡r✈❛❧ [ ✵ , ✶ ] ✇✐t❤ t❤❡ ♦♣❡r❛t✐♦♥s ⊕ ❛♥❞ ∗ ❞❡✜♥❡❞ s❡♠✐♠♦❞✉❧❡s ❜② x ⊕ y = min ( ✶ , x + y ) ❛♥❞ x ∗ = ✶ − x ✐s ❛♥ ❙❛r❛ ❱❛♥♥✉❝❝✐ ▼❱✲❛❧❣❡❜r❛✳ ❚❤✐s ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ✧t❤❡ st❛♥❞❛r❞ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ▼❱✲❛❧❣❡❜r❛✧ ❛♥❞ ✐t ✐s ✈❡r② s♣❡❝✐❛❧ ❜❡❝❛✉s❡ ✐t ❣❡♥❡r❛t❡s t❤❡ ✈❛r✐❡t② ♦❢ ▼❱✲❛❧❣❡❜r❛s✳ ✷ ❋♦r ❡✈❡r② n ∈ N ▲ n := { ✵ , ✶ / ( n − ✶ ) , · · · , ( n − ✷ ) / ( n − ✶ ) , ✶ } ②✐❡❧❞s ❛♥ ▼❱✲❛❧❣❡❜r❛ ✇✐t❤ t❤❡ ♦♣❡r❛t✐♦♥s ❞❡✜♥❡❞ ❛s t❤❡ r❡str✐❝t✐♦♥ ♦❢ t❤❡ st❛♥❞❛r❞ ▼❱✲❛❧❣❡❜r❛ ♦❢ t❤❡s❡ ♦♣❡r❛t✐♦♥s✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✸ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ ❖♥ ❡✈❡r② ▼❱✲❛❧❣❡❜r❛ A ✱ ✐t ✐s ♣♦ss✐❜❧❡ t♦ ❞❡✜♥❡ ❛♥♦t❤❡r s✈❛♥♥✉❝❝✐❅ ❝♦♥st❛♥t ✶ := ✵ ∗ ❛♥❞ t❤❡ ♦♣❡r❛t✐♦♥ x ⊙ y := ( x ∗ ⊕ y ∗ ) ∗ ✳ ❋r♦♠ ✉♥✐s❛✳✐t ♥♦✇ ♦♥ ✇❡ s❤❛❧❧ ✐♥❝❧✉❞❡ t❤❡s❡ ✐♥ t❤❡ s✐❣♥❛t✉r❡ ♦❢ t❤❡ ▼❱✲❛❧❣❡❜r❛✱ ❞❡♥♦t✐♥❣ A ✇✐t❤ ( A , ⊕ , ⊙ , ∗ , ✵ , ✶ ) ✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✹ ✴ ✷✶

❚❤❡ ◆❛t✉r❛❧ ♦r❞❡r ♦♥ ❛♥ ▼❱✲❆❧❣❡❜r❛ ❖♥ ■♥❥❡❝t✐✈✐t② ❋♦r ❛♥② ▼❱✲❛❧❣❡❜r❛ A ✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♦r❞❡r ❣✐✈❡♥ ❜②✿ ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ ⇒ x ∗ ⊕ y = ✶ , s❡♠✐♠♦❞✉❧❡s x ≤ y ⇐ ❙❛r❛ ❱❛♥♥✉❝❝✐ ❢♦r ❡✈❡r② x , y ∈ A ✳ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ❚❤❡ ♥❛t✉r❛❧ ♦r❞❡r ❞❡t❡r♠✐♥❡s ❛ str✉❝t✉r❡ ♦❢ ❜♦✉♥❞❡❞ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡ ♦♥ A ✱ ✇✐t❤ ✵ ❛♥❞ ✶ r❡s♣❡❝t✐✈❡❧② ❜♦tt♦♠ ❛♥❞ t♦♣ ❡❧❡♠❡♥t ❛♥❞ t❤❡ ♦♣❡r❛t✐♦♥s ♦❢ sup ❛♥❞ inf ❞❡✜♥❡❞ ❜②✿ x ∨ y := ( x ⊙ y ∗ ) ⊕ y ❛♥❞ x ∧ y := ( x ∗ ∨ y ∗ ) ∗ . ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✺ ✴ ✷✶

❚❤❡ ❇♦♦❧❡❛♥ ❈❡♥tr❡ ♦❢ ❛♥ ▼❱✲❛❧❣❡❜r❛ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❉❡✜♥✐t✐♦♥ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ▲❡t A ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✳ ❚❤❡ ❜♦♦❧❡❛♥ ❝❡♥tr❡ ♦❢ A B ( A ) ✐s t❤❡ ✉♥✐s❛✳✐t s❡t ♦❢ ❛❧❧ t❤❡ ❡❧❡♠❡♥ts x ∈ A s✉❝❤ t❤❛t x ⊕ x = x ✳ ❘❡♠❛r❦ B ( A ) ✐s ❛ s✉❜❛❧❣❡❜r❛ ♦❢ A ✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✻ ✴ ✷✶

❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ❛❜♦✉t ▼❱✲❛❧❣❡❜r❛s ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s Pr♦♣♦s✐t✐♦♥ ❙❛r❛ ▲❡t A ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✳ A ✐s ❝♦♠♣❧❡t❡ ❛♥❞ ✐ts ❜♦♦❧❡❛♥ ❝❡♥t❡r ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ B ( A ) ✐s ❛t♦♠✐❝ ✭❛s ❛ ❧❛tt✐❝❡✮ ✐✛ A ✐s ❛ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ ✉♥✐s❛✳✐t ❝♦♠♣❧❡t❡ ▼❱✲❝❤❛✐♥s✳ Pr♦♣♦s✐t✐♦♥ ❊✈❡r② ❝♦♠♣❧❡t❡ ▼❱✲❝❤❛✐♥ ✐s ❡✐t❤❡r ✐s♦♠♦r♣❤✐❝ t♦ ▲ n ❢♦r s♦♠❡ n ∈ N ♦r t♦ t❤❡ st❛♥❞❛r❞ ▼❱✲❛❧❣❡❜r❛ [ ✵ , ✶ ] ✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✼ ✴ ✷✶