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❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s

❙❛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 ✷✺✴✵✹✴✷✵✶✾ ✶ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

▼❱✲❆❧❣❡❜r❛s

❆r♦✉♥❞ ✶✾✺✽ ❈✳ ❈✳ ❈❤❛♥❣ ❞❡✈✐s❡❞ ▼❱✲❛❧❣❡❜r❛s t♦ st✉❞② ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✱ ✐♥ ♣❛rt✐❝✉❧❛r ▼❱✲❛❧❣❡❜r❛s ❢♦r♠ t❤❡ ❛❧❣❡❜r❛✐❝ s❡♠❛♥t✐❝s ♦❢ ❾✉❦❛s✐❡✇✐❝③ ❧♦❣✐❝✳ ❉❡✜♥✐t✐♦♥ ✭▼❱✲❛❧❣❡❜r❛✮ ❆♥ ▼❱✲❛❧❣❡❜r❛ ✐s ❛♥ ❛❧❣❡❜r❛ (A, ⊕,∗ , ✵) ♦❢ t②♣❡ ✭✷✱✶✱✵✮ s✉❝❤ 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 ✷✺✴✵✹✴✷✵✶✾ ✷ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❊①❛♠♣❧❡s ♦❢ ▼❱✲❛❧❣❡❜r❛s

❊①❛♠♣❧❡

✶ ❚❤❡ r❡❛❧ ✐♥t❡r✈❛❧ [✵, ✶] ✇✐t❤ t❤❡ ♦♣❡r❛t✐♦♥s ⊕ ❛♥❞ ∗ ❞❡✜♥❡❞

❜② x ⊕ y = min(✶, x + y) ❛♥❞ x∗ = ✶ − x ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳ ❚❤✐s ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ✧t❤❡ st❛♥❞❛r❞ ▼❱✲❛❧❣❡❜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❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

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

❚❤❡ ◆❛t✉r❛❧ ♦r❞❡r ♦♥ ❛♥ ▼❱✲❆❧❣❡❜r❛

❋♦r ❛♥② ▼❱✲❛❧❣❡❜r❛ A✱ t❤❡r❡ ❡①✐sts ❛ ♥❛t✉r❛❧ ♦r❞❡r ❣✐✈❡♥ ❜②✿ x ≤ y ⇐ ⇒ x∗ ⊕ y = ✶, ❢♦r ❡✈❡r② x, y ∈ A✳ ❚❤❡ ♥❛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 ✷✺✴✵✹✴✷✵✶✾ ✺ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❚❤❡ ❇♦♦❧❡❛♥ ❈❡♥tr❡ ♦❢ ❛♥ ▼❱✲❛❧❣❡❜r❛

❉❡✜♥✐t✐♦♥ ▲❡t A ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✳ ❚❤❡ ❜♦♦❧❡❛♥ ❝❡♥tr❡ ♦❢ A B(A) ✐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 ✷✺✴✵✹✴✷✵✶✾ ✻ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❙♦♠❡ ✉s❡❢✉❧ r❡s✉❧ts ❛❜♦✉t ▼❱✲❛❧❣❡❜r❛s

Pr♦♣♦s✐t✐♦♥ ▲❡t A ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✳ A ✐s ❝♦♠♣❧❡t❡ ❛♥❞ ✐ts ❜♦♦❧❡❛♥ ❝❡♥t❡r B(A) ✐s ❛t♦♠✐❝ ✭❛s ❛ ❧❛tt✐❝❡✮ ✐✛ A ✐s ❛ ❞✐r❡❝t ♣r♦❞✉❝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 ✷✺✴✵✹✴✷✵✶✾ ✼ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❆❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ❛♥❞ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐r✐♥❣s

❉❡✜♥✐t✐♦♥ ✭❆❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ❝♦♠♠✉t❛t✐✈❡ s❡♠✐r✐♥❣✮ ❆♥ ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ❝♦♠♠✉t❛t✐✈❡ s❡♠✐r✐♥❣ S ✐s ❛♥ ❛❧❣❡❜r❛ (S, +, ·, ✵, ✶) ♦❢ t②♣❡ ✭✷✱✷✱✵✱✵✮ s✉❝❤ t❤❛t✿

✶ (S, +, ✵) ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞❀ ✷ (S, ·, ✶) ✐s ❛ ♠♦♥♦✐❞❀ ✸ x · (y + z) = x · y + x · z ❛♥❞ (y + z) · x = y · x + z · x✱ ❢♦r

❡✈❡r② x, y, z ∈ S✳

✹ ✵ · s = ✵ = s · ✵✱ ❢♦r ❡❛❝❤ s ∈ S❀ ✺ s + s = s✱ ❢♦r ❡✈❡r② s ∈ S❀ ✻ x · y = y · x✱ ❢♦r ❡✈❡r② x, y ∈ S✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✽ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

▼❱✲s❡♠✐r✐♥❣s

❉❡✜♥✐t✐♦♥ ✭▼❱✲s❡♠✐r✐♥❣✮ ▲❡t S = (S, +, ·, ✵, ✶) ❜❡ ❛ ❝♦♠♠✉t❛t✐✈❡✱ ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t s❡♠✐r✐♥❣✳ S ✐s ❛ ▼❱✲s❡♠✐r✐♥❣ ✐✛ ❡①✐sts ❛ ♠❛♣ ∗ : S → S✱ ❝❛❧❧❡❞ ♥❡❣❛t✐♦♥✱ s✉❝❤ t❤❛t✿

✶ a · b = ✵ ✐✛ b ≤ a∗❀ ✷ a + b = (a∗ · (a∗ · b)∗)∗✳

❋r♦♠ ♥♦✇ ♦♥ ✇❡ s❤❛❧❧ ✐♥❝❧✉❞❡ t❤❡ ♥❡❣❛t✐♦♥ s②♠❜♦❧ ✐♥ t❤❡ s✐❣♥❛t✉r❡ ♦❢ t❤❡ ▼❱✲s❡♠✐r✐♥❣✱ ❞❡♥♦t✐♥❣ S ✇✐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❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

▼❱✲❆❧❣❡❜r❛s ❛♥❞ ▼❱✲s❡♠✐r✐♥❣s ❛r❡ ✐s♦♠♦r♣❤✐❝ ❝❛t❡❣♦r✐❡s

❚❤❡♦r❡♠ ▲❡t A ❜❡ ❛♥ ▼❱✲❛❧❣❡❜r❛✳ ❚❤❡♥ A∨⊙ = (A, ∨, ⊙, ✵, ✶) ❛♥❞ A∧⊕ = (A, ∧, ⊕, ✶, ✵) ❛r❡ s❡♠✐r✐♥❣s ❛♥❞ t❤❡ ✐♥✈♦❧✉t✐♦♥

∗ : A → A ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡♠✳ ■♥ ♣❛rt✐❝✉❧❛r A∨⊙

❛♥❞ A∧⊕ ❛r❡ ▼❱✲s❡♠✐r✐♥❣s ✇✐t❤ ♥❡❣❛t✐♦♥ ∗✳ ❚❤❡♦r❡♠ ■❢ (A, +, ·, ✵, ✶,∗ ) ✐s ❛♥ ▼❱✲s❡♠✐r✐♥❣✱ t❤❡ str✉❝t✉r❡ (A, ⊕, ·,∗ , ✵, ✶) ✇✐t❤✱ ❢♦r ❛❧❧ x, y ∈ A x ⊕ y = (x∗ · y∗)∗ ✐s ❛♥ ▼❱✲❛❧❣❡❜r❛✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❚❤❡ ♣r❡✈✐♦✉s ♣r♦♣♦s✐t✐♦♥s ❞❡✜♥❡ ❛ ❝♦✉♣❧❡ ♦❢ ❢✉♥❝t♦rs ✇❤✐❝❤ ♠❛❦❡ t❤❡ ❝❛t❡❣♦r✐❡s ♦❢ ▼❱✲❛❧❣❡❜r❛s ❛♥❞ ▼❱✲s❡♠✐r✐♥❣s ✐s♦♠♦r♣❤✐❝✳ ❚❤✐s ❛❧❧♦✇ ✉s t♦ ✐♠♣♦rt r❡s✉❧ts ❛♥❞ t❡❝❤♥✐q✉❡s ❢r♦♠ s❡♠✐r✐♥❣ ❛♥❞ r✐♥❣ t❤❡♦r② ✐♥ t❤❡ st✉❞② ♦❢ ▼❱✲❛❧❣❡❜r❛s✳ ❚❤❡ t❤❡♦r② ♦❢ ♠♦❞✉❧❡s ✭r❡s♣✱✳ s❡♠✐♠♦❞✉❧❡s✮ ✐s ❛♥ ❡ss❡♥t✐❛❧ ❝❤❛♣t❡r ✐♥ t❤❡ t❤❡♦r② ♦❢ r✐♥❣s ✭r❡s♣✱✳ s❡♠✐r✐♥❣s✮✳ ❲❡ ❢♦❝✉s❡❞ ♦♥ t❤❡ st✉❞② ♦❢ ✐♥❥❡❝t✐✈❡ ❛♥❞ ♣r♦❥❡❝t✐✈❡ ▼❱✲s❡♠✐♠♦❞✉❧❡s✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❙❡♠✐♠♦❞✉❧❡s

❉❡✜♥✐t✐♦♥ ✭❙❡♠✐♠♦❞✉❧❡✮ ▲❡t S ❜❡ ❛ s❡♠✐r✐♥❣✳ ❆ ✭❧❡❢t✮ S − semimodule ✐s ❛ ❝♦♠♠✉t❛t✐✈❡ ♠♦♥♦✐❞ (M, +, ✵) ✇✐t❤ ❛ s❝❛❧❛r ♠✉❧t✐♣❧✐❝❛t✐♦♥ · : (a, x) ∈ S × M → a · x ∈ M✱ s✉❝❤ t❤❛t t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❤♦❧❞ ❢♦r ❛❧❧ a, b ∈ S ❛♥❞ x, y ∈ M✿

✶ (ab) · x = a · (b · x)❀ ✷ a · (x + y) = (a · x) + (a · y)❀ ✸ (a + b) · x = (a · x) + (b · x)❀ ✹ ✵S · x = ✵M = a · ✵M❀ ✺ ✶ · x = x✳ ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t ✭❯♥✐✈❡rs✐tá ❞❡❣❧✐ ❙t✉❞✐ ❞✐ ❙❛❧❡r♥♦✮ ❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲s❡♠✐♠♦❞✉❧❡s ✷✺✴✵✹✴✷✵✶✾ ✶✷ ✴ ✷✶

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

■♥❥❡❝t✐✈❡ s❡♠✐♠♦❞✉❧❡s ✲ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts

❚❤❡♦r❡♠ ▲❡t S ❜❡ ❛♥ ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t s❡♠✐r✐♥❣ ❛♥❞ M ❛ ❧❡❢t S✲s❡♠✐♠♦❞✉❧❡✳ ❚❤❡♥✱ M ✐s ✐♥❥❡❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ t❤❡r❡ ❡①✐sts ❛ s❡t X s✉❝❤ t❤❛t M ✐s ❛ r❡tr❛❝t ♦❢ t❤❡ ❧❡❢t S✲s❡♠✐♠♦❞✉❧❡ HomB(S, B)X✱ ✇❤❡r❡ B ✐s t❤❡ ❇♦♦❧❡❛♥ s❡♠✐✜❡❧❞✳ Pr♦♣♦s✐t✐♦♥ ▲❡t S ❜❡ ❛♥ ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ❝♦♠♠✉t❛t✐✈❡ s❡♠✐r✐♥❣ ❛♥❞ M ❛♥ S✲s❡♠✐♠♦❞✉❧❡✳ ❚❤❡♥✱ M ✐s ✐♥❥❡❝t✐✈❡ ✐✛ t❤❡r❡ ❡①✐sts ❛ s❡t X s✉❝❤ t❤❛t M ✐s ❛ r❡tr❛❝t ♦❢ t❤❡ S✲s❡♠✐♠♦❞✉❧❡ Id(S)X ✭t❤❡ ✐❞❡❛❧s ♦❢ S s❡❡♥ ❛s ❛ ❥♦✐♥✲s❡♠✐❧❛tt✐❝❡✮✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

■♥❥❡❝t✐✈❡ s❡♠✐♠♦❞✉❧❡s ✲ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts

❙✐♥❝❡ ✇❡ ❛r❡ ✐♥t❡r❡st❡❞ ✐♥ ❝♦♠♠✉t❛t✐✈❡ s❡♠✐r✐♥❣s✱ ✇❡ ❝❛♥ ❦❡❡♣ ✐♥ ♠✐♥❞ t❤❡ s❡❝♦♥❞ ❝❤❛r❛❝t❡r✐③❛t✐♦♥ ♦❢ ✐♥❥❡❝t✐✈❡ s❡♠✐♠♦❞✉❧❡s t❤❛t s❡❡♠s ✭❛t ❧❡❛st t♦ ♠❡✮ ❡❛s✐❡r t♦ ❞❡❛❧ ✇✐t❤ ❛♥❞ ✇❡ ❝❛♥ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ ❈♦r♦❧❧❛r② ▲❡t S ❜❡ ❛♥ ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ✭❛♥❞ ❝♦♠♠✉t❛t✐✈❡✮ s❡♠✐r✐♥❣ ❛♥❞ M ❛♥ ✐♥❥❡❝t✐✈❡ ❧❡❢t S✲s❡♠✐♠♦❞✉❧❡✳ ❚❤❡♥ M ✐s ❛ ❝♦♠♣❧❡t❡ s❡♠✐♠♦❞✉❧❡✳ ■❢ ✐♥ ❛❞❞✐t✐♦♥ t❤❡ ❥♦✐♥✲s❡♠✐❧❛tt✐❝❡ S ✐s ❛ ❞✐str✐❜✉t✐✈❡ ❧❛tt✐❝❡✱ t❤❡♥ M ✐s ❛ ❝♦♠♣❧❡t❡ ❛♥❞ ✐♥✜♥✐t❡❧② ❞✐str✐❜✉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❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

Pr♦❥❡❝t✐✈❡ s❡♠✐♠♦❞✉❧❡s ✲ Pr❡❧✐♠✐♥❛r② r❡s✉❧ts

❆s r❡❣❛r❞s ♣r♦❥❡❝t✐✈❡ s❡♠✐♠♦❞✉❧❡s✱ ✇❡ ❤❛✈❡ t❤❛t ♣r♦❥❡❝t✐✈❡ ♦❜❥❡❝ts ❛r❡ t❤❡ r❡tr❛❝ts ♦❢ t❤❡ ❢r❡❡ ♦♥❡s✳ ●✐✈❡♥ ❛ s❡♠✐r✐♥❣ S ❛♥❞ ❛ s❡t X ✇❡ ❦♥♦✇ t❤❛t t❤❡ ❢r❡❡ S✲s❡♠✐♠♦❞✉❧❡ ❣❡♥❡r❛t❡❞ ❜② X ✐s t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s ❢r♦♠ X t♦ S ✇✐t❤ ✜♥✐t❡ s✉♣♣♦rt t❤❛t ✇❡ ❞❡♥♦t❡ ✇✐t❤ S(X)✳ ❙♦✱ ✇❡ ❤❛✈❡ t❤❡ ❢♦❧❧♦✇✐♥❣ Pr♦♣♦s✐t✐♦♥ ▲❡t S ❜❡ ❛ s❡♠✐r✐♥❣ ❛♥❞ M ❛ S✲s❡♠✐♠♦❞✉❧❡✳ ❚❤❡♥ M ✐s ♣r♦❥❡❝t✐✈❡ ✐✛ ✐t ✐s ❛ r❡tr❛❝t ♦❢ S(X) ❢♦r s♦♠❡ s❡t X✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

▲❡♠♠❛ ✭✶✮ ❋♦r ❡❛❝❤ ✐♥t❡❣❡r n ≥ ✷✱ ▲n ∼ = Id(▲n) ❛s ▲n✲s❡♠✐♠♦❞✉❧❡s✳ ❈♦♥s❡q✉❡♥t❧②✱ ▲n ✐s ❛ s❡❧❢✲✐♥❥❡❝t✐✈❡ s❡♠✐r✐♥❣✳ ✭✷✮ [✵, ✶] ✐s ❛ s❡❧❢✲✐♥❥❡❝t✐✈❡ s❡♠✐r✐♥❣✳ ▲❡♠♠❛ ▲❡t S =

i∈I Si ❜❡ ❛ ❞✐r❡❝t ♣r♦❞✉❝t ♦❢ s❡♠✐r✐♥❣s Si✳ ❚❤❡♥ S ✐s

❧❡❢t s❡❧❢✲✐♥❥❡❝t✐✈❡ ✐❢ ❛♥❞ ♦♥❧② ✐❢ ❡❛❝❤ Si ✐s ❧❡❢t s❡❧❢✲✐♥❥❡❝t✐✈❡✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❚❤❡♦r❡♠ ❋♦r ❛♥② ▼❱✲❛❧❣❡❜r❛ A ✇✐t❤ ❛♥ ❛t♦♠✐❝ ❇♦♦❧❡❛♥ ❝❡♥t❡r✱ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥❞✐t✐♦♥s ❛r❡ ❡q✉✐✈❛❧❡♥t✿ ✭✶✮ ❚❤❡ s❡♠✐r✐♥❣ A∨⊙ ✐s s❡❧❢✲✐♥❥❡❝t✐✈❡❀ ✭✷✮ ❆❧❧ ✜♥✐t❡❧② ❣❡♥❡r❛t❡❞ ♣r♦❥❡❝t✐✈❡ A∨⊙✲s❡♠✐♠♦❞✉❧❡s ❛r❡ ✐♥❥❡❝t✐✈❡❀ ✭✸✮ A ✐s ❛ ❝♦♠♣❧❡t❡ ▼❱✲❛❧❣❡❜r❛✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❘❡❢❡r❡♥❝❡s

✶ ❏✳ ❙✳ ●♦❧❛♥✱ ❙❡♠✐r✐♥❣s ❛♥❞ t❤❡✐r ❆♣♣❧✐❝❛t✐♦♥s✱ ❑❧✉✇❡r

❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ❉♦r❞r❡❝❤t✲❇♦st♦♥✲▲♦♥❞♦♥✱ ✶✾✾✾✳

✷ ❘✳ ▲✳ ❖✳ ❈✐❣♥♦❧✐✱ ■✳ ▼✳ ▲✳ ❉✬❖tt❛✈✐❛♥♦ ❛♥❞ ❉✳ ▼✉♥❞✐❝✐✱

❆❧❣❡❜r❛✐❝ ❢♦✉♥❞❛t✐♦♥s ♦❢ ♠❛♥②✲✈❛❧✉❡❞ r❡❛s♦♥✐♥❣✳ ❚r❡♥❞s ✐♥ ▲♦❣✐❝✳ ❱♦❧✳ ✼✳ ❉♦r❞r❡❝❤t✿ ❑❧✉✇❡r ❆❝❛❞❡♠✐❝ P✉❜❧✐s❤❡rs✱ ✷✵✵✵✳

✸ ❆✳ ❉✐ ◆♦❧❛ ❛♥❞ ❇✳ ●❡r❧❛✱ ❆❧❣❡❜r❛s ♦❢ ❾✉❦❛s✐❡✇✐❝③✬s ❧♦❣✐❝

❛♥❞ t❤❡✐r s❡♠✐r✐♥❣ r❡❞✉❝ts✱ ■❞❡♠♣♦t❡♥t ▼❛t❤❡♠❛t✐❝s ❛♥❞ ▼❛t❤❡♠❛t✐❝❛❧ P❤②s✐❝s✱ ✐♥✿ ❈♦♥t❡♠♣✳ ▼❛t❤✳✱ ❱♦❧✳ ✸✼✼✱ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✷✵✵✺✱ ♣♣✳ ✶✸✹✕✶✺✹✳

✹ ❆✳ ❉✐ ◆♦❧❛ ❛♥❞ ❈✳ ❘✉ss♦✱ ❙❡♠✐r✐♥❣ ❛♥❞ s❡♠✐♠♦❞✉❧❡ ✐ss✉❡s

✐♥ ▼❱✲❛❧❣❡❜r❛s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✹✶ ✭✷✵✶✸✮✱ ✶✵✶✼✕✶✵✹✽✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❘❡❢❡r❡♥❝❡s

✶ ❆✳ ❉✐ ◆♦❧❛✱ ●✳ ▲❡♥③✐✱ ❚✳ ●✳ ◆❛♠ ❛♥❞ ❙✳ ❱❛♥♥✉❝❝✐✱ ❖♥

✐♥❥❡❝t✐✈✐t② ♦❢ s❡♠✐♠♦❞✉❧❡s ♦✈❡r ❛❞❞✐t✐✈❡❧② ✐❞❡♠♣♦t❡♥t ❞✐✈✐s✐♦♥ s❡♠✐r✐♥❣s ❛♥❞ ❝❤❛✐♥ ▼❱✲s❡♠✐r✐♥❣s✱ ❛❝❝❡♣t❡❞ ❢♦r ♣✉❜❧✐❝❛t✐♦♥✳

✷ ●✳ ●rät③❡r✱ ●❡♥❡r❛❧ ▲❛tt✐❝❡ ❚❤❡♦r②✱ ✷♥❞ ❊❞✱ ❇✐r❦❤ä✉s❡r

❱❡r❧❛❣✱ ❇❛s❡❧✲❇♦st♦♥✲❇❡r❧✐♥✱ ✶✾✾✽✳

✸ ❏✳ ❆❞á♠❡❦✱ ❍✳ ❍❡rr❧✐❝❤✱ ●✳ ❊✳ ❙tr❡❝❦❡r✱ ❆❜str❛❝t ❛♥❞

❈♦♥❝r❡t❡ ❈❛t❡❣♦r✐❡s ✲ ❚❤❡ ❏♦② ♦❢ ❈❛ts ✭✷✵✵✾✮✳

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

❖♥ ■♥❥❡❝t✐✈✐t② ❛♥❞ Pr♦❥❡❝t✐✈✐t② ♦❢ ▼❱✲ s❡♠✐♠♦❞✉❧❡s ❙❛r❛ ❱❛♥♥✉❝❝✐ s✈❛♥♥✉❝❝✐❅ ✉♥✐s❛✳✐t

❚❍❆◆❑ ❨❖❯ ❋❖❘ ❨❖❯❘ ❆❚❚❊◆❚■❖◆✦

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