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❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✲ ❥♦✐♥t ✇♦r❦ ✇✐t❤ ▼❛♥✉❡❧ ❇♦❞✐rs❦②

▲■❳ ✲ ❊❝♦❧❡ P♦❧②t❡❝❤♥✐q✉❡

❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶ ✴ ✷✹

❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠s

■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ ❈❙Ps

❆ ❈❙P ✐s ❛ ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠✳ ❚❤❡ ✐♥♣✉t ❝♦♥s✐sts ♦❢ ❛ ✜♥✐t❡ s❡t ♦❢ ✈❛r✐❛❜❧❡s ❛♥❞ ❛ ✜♥✐t❡ s❡t ♦❢ ❝♦♥str❛✐♥ts ✐♠♣♦s❡❞ ♦♥ t❤♦s❡ ✈❛r✐❛❜❧❡s✳ ❚❤❡ t❛s❦ ✐s t♦ ❞❡❝✐❞❡ ✇❤❡t❤❡r t❤❡r❡ ✐s ❛♥ ❛ss✐❣♥♠❡♥t ♦❢ ✈❛❧✉❡s t♦ t❤❡ ✈❛r✐❛❜❧❡s s✉❝❤ t❤❛t ❛❧❧ t❤❡ ❝♦♥str❛✐♥ts ❛r❡ s✐♠✉❧t❛♥❡♦✉s❧② s❛t✐s✜❡❞✳

❊①❛♠♣❧❡s

■s ❛ ♣r♦♣♦s✐t✐♦♥❛❧ ❢♦r♠✉❧❛ ✐♥ ❈◆❋ ✇✐t❤ ❛t ♠♦st t❤r❡❡ ❧✐t❡r❛❧s ♣❡r ❝❧❛✉s❡ s❛t✐s✜❛❜❧❡ ♦♥ {✵, ✶}❄ ■s t❤❡r❡ ❛ s♦❧✉t✐♦♥ t♦ ❛ ✜♥✐t❡ s❡t ♦❢ ❧✐♥❡❛r ❡q✉❛t✐♦♥s ♦✈❡r F✷❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷ ✴ ✷✹

❋♦r♠❛❧✐s❛t✐♦♥ ♦❢ CSPs

Pr❡❧✐♠✐♥❛r✐❡s

• ✐✈❡♥ ❛ s✐❣♥❛t✉r❡ τ✱ ❛♥ ❛t♦♠✐❝ ❢♦r♠✉❧❛ ✐s ♦❢ t❤❡ ❢♦r♠ R(x) ✇✐t❤ R ❛

r❡❧❛t✐♦♥ ✐♥ τ✳ ❆ ♣r✐♠✐t✐✈❡ ♣♦s✐t✐✈❡ ❢♦r♠✉❧❛ ♦♥ ❛ s✐❣♥❛t✉r❡ τ ✐s ♦❢ t❤❡ ❢♦r♠ ∃x✶ . . . xn(φ✶(x) ∧ · · · ∧ φk(x)) ✇❤❡r❡ ❛❧❧ φi ❛r❡ ❛t♦♠✐❝ ❢♦r♠✉❧❛s✳ ❆♥ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ❛ τ✲str✉❝t✉r❡ Γ t♦ ❛ τ✲str✉❝t✉r❡ ∆ ✐s ❛ ♠❛♣ ✇❤✐❝❤ ♣r❡s❡r✈❡s ❛❧❧ ❛t♦♠✐❝ ❢♦r♠✉❧❛s✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✸ ✴ ✷✹

❋♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ CSPs

• ✐✈❡♥ ❛ str✉❝t✉r❡ Γ ♦♥ ❛ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ s✐❣♥❛t✉r❡ τ✱ ✇❡ ❞❡✜♥❡ t❤❡

❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❈❙P(Γ)✿

⋄ ■♥♣✉t✿ ❛ ♣r✐♠✐t✐✈❡ ♣♦s✐t✐✈❡ s❡♥t❡♥❝❡ φ✳ ⋄ ◗✉❡st✐♦♥✿ Γ | = φ ❄

❊q✉✐✈❛❧❡♥t❧②✿

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ τ✲str✉❝t✉r❡ ∆✳ ⋄ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ∆ t♦ Γ❄

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ str✉❝t✉r❡ ❝❛♥ ❜❡ ❡✐t❤❡r ✜♥✐t❡✱ ♦r ✐♥✜♥✐t❡✳ ▼❛♥② ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❈❙Ps✳ ◆❛t✉r❛❧ q✉❡st✐♦♥✿ ✇❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P ❢♦r ❛ ❣✐✈❡♥ ❄

■♠♣♦rt❛♥t ♣r♦♣❡rt②

❋♦r ❛♥② r❡❧❛t✐♦♥ ♣♣✲❞❡✜♥❛❜❧❡ ♦♥ ❛ str✉❝t✉r❡ ✱ ❈❙P ❛♥❞ ❈❙P ❛r❡ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❡q✉✐✈❛❧❡♥t✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✹ ✴ ✷✹

❋♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ CSPs

• ✐✈❡♥ ❛ str✉❝t✉r❡ Γ ♦♥ ❛ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ s✐❣♥❛t✉r❡ τ✱ ✇❡ ❞❡✜♥❡ t❤❡

❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❈❙P(Γ)✿

⋄ ■♥♣✉t✿ ❛ ♣r✐♠✐t✐✈❡ ♣♦s✐t✐✈❡ s❡♥t❡♥❝❡ φ✳ ⋄ ◗✉❡st✐♦♥✿ Γ | = φ ❄

❊q✉✐✈❛❧❡♥t❧②✿

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ τ✲str✉❝t✉r❡ ∆✳ ⋄ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ∆ t♦ Γ❄

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ str✉❝t✉r❡ Γ ❝❛♥ ❜❡ ❡✐t❤❡r ✜♥✐t❡✱ ♦r ✐♥✜♥✐t❡✳ ▼❛♥② ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❈❙Ps✳ ◆❛t✉r❛❧ q✉❡st✐♦♥✿ ✇❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P(Γ) ❢♦r ❛ ❣✐✈❡♥ Γ❄

■♠♣♦rt❛♥t ♣r♦♣❡rt②

❋♦r ❛♥② r❡❧❛t✐♦♥ ♣♣✲❞❡✜♥❛❜❧❡ ♦♥ ❛ str✉❝t✉r❡ ✱ ❈❙P ❛♥❞ ❈❙P ❛r❡ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❡q✉✐✈❛❧❡♥t✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✹ ✴ ✷✹

❋♦r♠❛❧ ❞❡✜♥✐t✐♦♥ ♦❢ CSPs

• ✐✈❡♥ ❛ str✉❝t✉r❡ Γ ♦♥ ❛ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ s✐❣♥❛t✉r❡ τ✱ ✇❡ ❞❡✜♥❡ t❤❡

❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠ ❈❙P(Γ)✿

⋄ ■♥♣✉t✿ ❛ ♣r✐♠✐t✐✈❡ ♣♦s✐t✐✈❡ s❡♥t❡♥❝❡ φ✳ ⋄ ◗✉❡st✐♦♥✿ Γ | = φ ❄

❊q✉✐✈❛❧❡♥t❧②✿

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ τ✲str✉❝t✉r❡ ∆✳ ⋄ ◗✉❡st✐♦♥✿ ■s t❤❡r❡ ❛♥ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ ∆ t♦ Γ❄

❚❤❡ ❞♦♠❛✐♥ ♦❢ t❤❡ str✉❝t✉r❡ Γ ❝❛♥ ❜❡ ❡✐t❤❡r ✜♥✐t❡✱ ♦r ✐♥✜♥✐t❡✳ ▼❛♥② ❝♦♠♣✉t❛t✐♦♥❛❧ ♣r♦❜❧❡♠s ❝❛♥ ❜❡ ♠♦❞❡❧❧❡❞ ❛s ❈❙Ps✳ ◆❛t✉r❛❧ q✉❡st✐♦♥✿ ✇❤❛t ✐s t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P(Γ) ❢♦r ❛ ❣✐✈❡♥ Γ❄

■♠♣♦rt❛♥t ♣r♦♣❡rt②

❋♦r ❛♥② r❡❧❛t✐♦♥ R ♣♣✲❞❡✜♥❛❜❧❡ ♦♥ ❛ str✉❝t✉r❡ Γ✱ ❈❙P(Γ) ❛♥❞ ❈❙P(Γ, R) ❛r❡ ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❡q✉✐✈❛❧❡♥t✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✹ ✴ ✷✹

❊①❛♠♣❧❡✿ ✸✲❝♦❧♦r❛❜✐❧✐t②

❚❤❡ ❢❛♠♦✉s ♣r♦❜❧❡♠✳✳✳ ✸✲❝♦❧♦r❛❜✐❧✐t②

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ ❣r❛♣❤ G✳ ⋄ ◗✉❡st✐♦♥✿ ❝❛♥ t❤❡ ✈❡rt✐❝❡s ❜❡ ❝♦❧♦r❡❞ ✇✐t❤ t❤r❡❡ ❝♦❧♦rs s✉❝❤ t❤❛t ♥♦ t✇♦ ✈❡rt✐❝❡s ❛❞❥❛❝❡♥t ✐♥ G ❣❡t t❤❡ s❛♠❡ ❝♦❧♦r❄

✳✳✳ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❛ ❈❙P ✇✐t❤ ❛ ✜♥✐t❡ str✉❝t✉r❡✿ ❈❙P(K✸)✿

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ ❣r❛♣❤ G✳ ⋄ ◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ K✸❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✺ ✴ ✷✹

❊①❛♠♣❧❡✿ ❉✐r❡❝t❡❞ ●r❛♣❤ ❆❝②❝❧✐❝✐t②

❚❤❡ ❢❛♠♦✉s ♣r♦❜❧❡♠✳✳✳ ❉✐r❡❝t❡❞ ●r❛♣❤ ❆❝②❝❧✐❝✐t②

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ ❞✐r❡❝t❡❞ ❣r❛♣❤ G✳ ⋄ ◗✉❡st✐♦♥✿ ✐s G ❛❝②❝❧✐❝❄

✳✳✳ ❝❛♥ ❜❡ ❢♦r♠✉❧❛t❡❞ ❛s ❛ ❈❙P ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ str✉❝t✉r❡✿ ❈❙P(Q, <)✿

⋄ ■♥♣✉t✿ ❛ ✜♥✐t❡ ❞✐r❡❝t❡❞ ❣r❛♣❤ G✳ ⋄ ◗✉❡st✐♦♥✿ ✐s t❤❡r❡ ❛ ❤♦♠♦♠♦r♣❤✐s♠ ❢r♦♠ G t♦ (Q, <)❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✻ ✴ ✷✹

❉✐❝❤♦t♦♠② ❢♦r ✜♥✐t❡ ❙tr✉❝t✉r❡s

❙❝❤❛❡❢❡r✬✼✼✿ ❢♦r ❛♥② ✷✲❡❧❡♠❡♥t str✉❝t✉r❡ Γ✱ ❈❙P(Γ) ✐s ❡✐t❤❡r ♣♦❧②♥♦♠✐❛❧❧② s♦❧✈❛❜❧❡ ♦r ◆P✲❝♦♠♣❧❡t❡✳

❈♦♥❥❡❝t✉r❡

❋❡❞❡r✲❱❛r❞✐✬✾✸ ✭❝♦♥❥❡❝t✉r❡✮✿ t❤✐s ❞✐❝❤♦t♦♠② ❤♦❧❞s ❢♦r ❡✈❡r② ✜♥✐t❡ str✉❝t✉r❡ Γ✳ ❇✉❧❛t♦✈✬✵✸✿ ❝♦♥✜r♠❡❞ ❋❡❞❡r✲❱❛r❞✐✬s ❝♦♥❥❡❝t✉r❡ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✸✳ ▼❛r❦♦✈✐❝✬✶✷✿ ❝♦♥✜r♠❡❞ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✹ ✭❛♥♥♦✉♥❝❡❞ ❜✉t ♥♦t ♣✉❜❧✐s❤❡❞ ②❡t✮✳ ❚❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ❛❧r❡❛❞② ♦♣❡♥ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ≥ ✺✳ ❲❤❛t ❛❜♦✉t ✐♥✜♥✐t❡ str✉❝t✉r❡s❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✼ ✴ ✷✹

❉✐❝❤♦t♦♠② ❢♦r ✜♥✐t❡ ❙tr✉❝t✉r❡s

❙❝❤❛❡❢❡r✬✼✼✿ ❢♦r ❛♥② ✷✲❡❧❡♠❡♥t str✉❝t✉r❡ Γ✱ ❈❙P(Γ) ✐s ❡✐t❤❡r ♣♦❧②♥♦♠✐❛❧❧② s♦❧✈❛❜❧❡ ♦r ◆P✲❝♦♠♣❧❡t❡✳

❈♦♥❥❡❝t✉r❡

❋❡❞❡r✲❱❛r❞✐✬✾✸ ✭❝♦♥❥❡❝t✉r❡✮✿ t❤✐s ❞✐❝❤♦t♦♠② ❤♦❧❞s ❢♦r ❡✈❡r② ✜♥✐t❡ str✉❝t✉r❡ Γ✳ ❇✉❧❛t♦✈✬✵✸✿ ❝♦♥✜r♠❡❞ ❋❡❞❡r✲❱❛r❞✐✬s ❝♦♥❥❡❝t✉r❡ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✸✳ ▼❛r❦♦✈✐❝✬✶✷✿ ❝♦♥✜r♠❡❞ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✹ ✭❛♥♥♦✉♥❝❡❞ ❜✉t ♥♦t ♣✉❜❧✐s❤❡❞ ②❡t✮✳ ❚❤❡ ❝♦♥❥❡❝t✉r❡ ✐s ❛❧r❡❛❞② ♦♣❡♥ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ≥ ✺✳ ❲❤❛t ❛❜♦✉t ✐♥✜♥✐t❡ str✉❝t✉r❡s❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✼ ✴ ✷✹

■♥✜♥✐t❡ ❙tr✉❝t✉r❡s

◆♦♥✲❉✐❝❤♦t♦♠②

❇♦❞✐rs❦②✲●r♦❤❡✬✵✽✿ ❊✈❡r② ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❈❙P ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ t❡♠♣❧❛t❡✳ ▲❛❞♥❡r✬✼✺✿ ✐❢ P = ◆P✱ t❤❡r❡ ❛r❡ ◆P✲✐♥t❡r♠❡❞✐❛t❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✱ ✐✳❡✳✱ ♣r♦❜❧❡♠s ✐♥ ◆P t❤❛t ❛r❡ ♥❡✐t❤❡r ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ tr❛❝t❛❜❧❡ ♥♦r ◆P✲❝♦♠♣❧❡t❡✳ ❈♦♥s❡q✉❡♥t❧②✿ ♥♦ ❞✐❝❤♦t♦♠② ❢♦r ❈❙Ps ♦♥ ✐♥✜♥✐t❡ str✉❝t✉r❡s✳

◗✉❡st✐♦♥

❈❛♥ ✇❡ ✐❞❡♥t✐❢② ❧❛r❣❡ ♥❛t✉r❛❧ ❝❧❛ss❡s ♦❢ ❈❙Ps ♦♥ ✐♥✜♥✐t❡ str✉❝t✉r❡s ✇❤♦s❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ❝❧❛ss✐✜❡❞❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✽ ✴ ✷✹

■♥✜♥✐t❡ ❙tr✉❝t✉r❡s

◆♦♥✲❉✐❝❤♦t♦♠②

❇♦❞✐rs❦②✲●r♦❤❡✬✵✽✿ ❊✈❡r② ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠ ✐s ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ ❡q✉✐✈❛❧❡♥t t♦ ❛ ❈❙P ✇✐t❤ ❛♥ ✐♥✜♥✐t❡ t❡♠♣❧❛t❡✳ ▲❛❞♥❡r✬✼✺✿ ✐❢ P = ◆P✱ t❤❡r❡ ❛r❡ ◆P✲✐♥t❡r♠❡❞✐❛t❡ ❝♦♠♣✉t❛t✐♦♥❛❧ ❞❡❝✐s✐♦♥ ♣r♦❜❧❡♠s✱ ✐✳❡✳✱ ♣r♦❜❧❡♠s ✐♥ ◆P t❤❛t ❛r❡ ♥❡✐t❤❡r ♣♦❧②♥♦♠✐❛❧✲t✐♠❡ tr❛❝t❛❜❧❡ ♥♦r ◆P✲❝♦♠♣❧❡t❡✳ ❈♦♥s❡q✉❡♥t❧②✿ ♥♦ ❞✐❝❤♦t♦♠② ❢♦r ❈❙Ps ♦♥ ✐♥✜♥✐t❡ str✉❝t✉r❡s✳

◗✉❡st✐♦♥

❈❛♥ ✇❡ ✐❞❡♥t✐❢② ❧❛r❣❡ ♥❛t✉r❛❧ ❝❧❛ss❡s ♦❢ ❈❙Ps ♦♥ ✐♥✜♥✐t❡ str✉❝t✉r❡s ✇❤♦s❡ ❝♦♠♣❧❡①✐t② ❝❛♥ ❜❡ ❝❧❛ss✐✜❡❞❄

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✽ ✴ ✷✹

▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ❈❙Ps✿ ❲❡ ❞❡♥♦t❡ ❜② P(N) t❤❡ ♣♦✇❡r s❡t ♦❢ N✳ ❈❙P(P(N), x ∩ y ⊆ z, x y, x y) ✇❤❡r❡ st❛t❡s ❢♦r ❞✐s❥♦✐♥t♥❡ss✳ ⋄ ❊①✳ ♦❢ ✐♥♣✉t✿ (x ∩ y ⊆ z) ∧ (t z) ∧ (t x) ∧ (t ∩ z ⊆ y) ⋄ ❆♥s✇❡r✿ ②❡s✱ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✳ ❈❙P ❊①✳ ♦❢ ✐♥♣✉t✿ ❆♥s✇❡r✿ ◆♦ s♦❧✉t✐♦♥✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✾ ✴ ✷✹

▼♦r❡ ❡①❛♠♣❧❡s ♦❢ ❈❙Ps✿ ❲❡ ❞❡♥♦t❡ ❜② P(N) t❤❡ ♣♦✇❡r s❡t ♦❢ N✳ ❈❙P(P(N), x ∩ y ⊆ z, x y, x y) ✇❤❡r❡ st❛t❡s ❢♦r ❞✐s❥♦✐♥t♥❡ss✳ ⋄ ❊①✳ ♦❢ ✐♥♣✉t✿ (x ∩ y ⊆ z) ∧ (t z) ∧ (t x) ∧ (t ∩ z ⊆ y) ⋄ ❆♥s✇❡r✿ ②❡s✱ t❤❡r❡ ✐s ❛ s♦❧✉t✐♦♥✳ ❈❙P(P(N), x ∩ y = z, x ∪ y = z, x = y) ⋄ ❊①✳ ♦❢ ✐♥♣✉t✿ (x ∩ y = z) ∧ (x ∪ y = t) ∧ (t ∩ u = z) ∧ (u ∩ x = x) ∧ (u = x) ⋄ ❆♥s✇❡r✿ ◆♦ s♦❧✉t✐♦♥✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✾ ✴ ✷✹

• ❡♥❡r❛❧✐t✐❡s ♦♥ s❡t ❈❙Ps

❚❤❡s❡ ♣r♦❜❧❡♠s ❜❡❧♦♥❣ t♦ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❈❙Ps ❝❛❧❧❡❞ s❡t ❈❙Ps✳

■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥

❆ s❡t ❈❙P ✐s ❛ ❈❙P ✇❤❡r❡ t❤❡ ✈❛r✐❛❜❧❡s t❛❦❡ ❛s ✈❛❧✉❡s s✉❜s❡ts ♦❢ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ■t ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ s❡t ♦❢ ✧❜♦♦❧❡❛♥ ❝♦♥str❛✐♥ts✧ ❛❧❧♦✇❡❞ ✐♥ t❤❡ ✐♥♣✉t✳ ❙❡t ❈❙Ps ❛♣♣❡❛r ♥❛t✉r❛❧❧② ✐♥ ✈❛r✐♦✉s ❛r❡❛s ❧✐❦❡ ♣r♦❣r❛♠ ❛♥❛❧②s✐s ✭❑✉♥❝❛❦✲◆❣✉②❡♥✲❘✐♥❛r❞✬✵✻✮✱ ❦♥♦✇❧❡❞❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭❑üst❡rs✲▼♦❧✐t♦r✬✵✷✮✱ ❛♥❞ s♣❛t✐❛❧ r❡❛s♦♥✐♥❣ ✭❉r❛❦❡♥❣r❡♥✲❏♦♥ss♦♥✬✾✼✮✳ ▼❛r✐♦tt✲❖❞❡rs❦②✬✾✻✿ ❛❧❧ s❡t ❈❙Ps ❛r❡ ✐♥ ◆P✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✵ ✴ ✷✹

• ❡♥❡r❛❧✐t✐❡s ♦♥ s❡t ❈❙Ps

❚❤❡s❡ ♣r♦❜❧❡♠s ❜❡❧♦♥❣ t♦ ❛ ❧❛r❣❡ ❝❧❛ss ♦❢ ❈❙Ps ❝❛❧❧❡❞ s❡t ❈❙Ps✳

■♥❢♦r♠❛❧ ❞❡✜♥✐t✐♦♥

❆ s❡t ❈❙P ✐s ❛ ❈❙P ✇❤❡r❡ t❤❡ ✈❛r✐❛❜❧❡s t❛❦❡ ❛s ✈❛❧✉❡s s✉❜s❡ts ♦❢ t❤❡ ♥❛t✉r❛❧ ♥✉♠❜❡rs✳ ■t ✐s ♣❛r❛♠❡tr✐③❡❞ ❜② t❤❡ s❡t ♦❢ ✧❜♦♦❧❡❛♥ ❝♦♥str❛✐♥ts✧ ❛❧❧♦✇❡❞ ✐♥ t❤❡ ✐♥♣✉t✳ ❙❡t ❈❙Ps ❛♣♣❡❛r ♥❛t✉r❛❧❧② ✐♥ ✈❛r✐♦✉s ❛r❡❛s ❧✐❦❡ ♣r♦❣r❛♠ ❛♥❛❧②s✐s ✭❑✉♥❝❛❦✲◆❣✉②❡♥✲❘✐♥❛r❞✬✵✻✮✱ ❦♥♦✇❧❡❞❣❡ r❡♣r❡s❡♥t❛t✐♦♥ ✭❑üst❡rs✲▼♦❧✐t♦r✬✵✷✮✱ ❛♥❞ s♣❛t✐❛❧ r❡❛s♦♥✐♥❣ ✭❉r❛❦❡♥❣r❡♥✲❏♦♥ss♦♥✬✾✼✮✳ ▼❛r✐♦tt✲❖❞❡rs❦②✬✾✻✿ ❛❧❧ s❡t ❈❙Ps ❛r❡ ✐♥ ◆P✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✵ ✴ ✷✹

❉❡✜♥✐t✐♦♥ ♦❢ ❛ s❡t ❈❙P

❘❡❞✉❝ts

❆ r❡❞✉❝t Γ ♦❢ ❛ str✉❝t✉r❡ ∆ ✐s ❛ str✉❝t✉r❡ ✇✐t❤ ❛ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ s✐❣♥❛t✉r❡ ❛♥❞ ✇❤♦s❡ r❡❧❛t✐♦♥s ❛r❡ ❞❡✜♥❛❜❧❡ ♦✈❡r ∆ ❜② ❛ q✉❛♥t✐✜❡r ❢r❡❡ ❢♦r♠✉❧❛✳

❊①❛♠♣❧❡s

▲❡t B = (P(N), ∩, ∪, ∁, ∅, N) ❜❡ t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ t❤❡ s✉❜s❡ts ♦❢ N✳ (P(N), {(x, y) | x ∩ y = x}, {(x, y) | x ∩ y = ∅}) ✐s ❛ r❡❞✉❝t ♦❢ B✳ (P(N), {(x, y, z) | (x ∪ y) ∩ (∁x ∪ ∁y) = z} ❛s ✇❡❧❧✳

❙❡t ❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠s

❆ s❡t ❈❙P ✐s ❛ ❈❙P ❢♦r ❛ r❡❞✉❝t ♦❢ ✳

❊①❛♠♣❧❡s

❈❙P ❛♥❞ ❈❙P ❛r❡ s❡t ❈❙Ps✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✶ ✴ ✷✹

❉❡✜♥✐t✐♦♥ ♦❢ ❛ s❡t ❈❙P

❘❡❞✉❝ts

❆ r❡❞✉❝t Γ ♦❢ ❛ str✉❝t✉r❡ ∆ ✐s ❛ str✉❝t✉r❡ ✇✐t❤ ❛ ✜♥✐t❡ r❡❧❛t✐♦♥❛❧ s✐❣♥❛t✉r❡ ❛♥❞ ✇❤♦s❡ r❡❧❛t✐♦♥s ❛r❡ ❞❡✜♥❛❜❧❡ ♦✈❡r ∆ ❜② ❛ q✉❛♥t✐✜❡r ❢r❡❡ ❢♦r♠✉❧❛✳

❊①❛♠♣❧❡s

▲❡t B = (P(N), ∩, ∪, ∁, ∅, N) ❜❡ t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ♦❢ t❤❡ s✉❜s❡ts ♦❢ N✳ (P(N), {(x, y) | x ∩ y = x}, {(x, y) | x ∩ y = ∅}) ✐s ❛ r❡❞✉❝t ♦❢ B✳ (P(N), {(x, y, z) | (x ∪ y) ∩ (∁x ∪ ∁y) = z} ❛s ✇❡❧❧✳

❙❡t ❈♦♥str❛✐♥t ❙❛t✐s❢❛❝t✐♦♥ Pr♦❜❧❡♠s

❆ s❡t ❈❙P ✐s ❛ ❈❙P ❢♦r ❛ r❡❞✉❝t ♦❢ (P(N), ∩, ∪, ∁, ∅, N)✳

❊①❛♠♣❧❡s

❈❙P(P(N), , , =) ❛♥❞ ❈❙P(P(N), ⊖, =) ❛r❡ s❡t ❈❙Ps✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✶ ✴ ✷✹

❈♦♥❥❡❝t✉r❡

❊✈❡r② s❡t ❈❙P ✐s ❡✐t❤❡r ✐♥ P✱ ♦r ◆P✲❝♦♠♣❧❡t❡✳

❯♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤

❋♦r ✜♥✐t❡ str✉❝t✉r❡s✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙Ps ❝❛♥ ❜❡ ❛♥❛❧②s❡❞ ✉s✐♥❣ ❛ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤✳ ❚❤✐s ❛♣♣r♦❛❝❤ ✇❛s ✉s❡❞ ❜② ❇✉❧❛t♦✈✬✵✸ t♦ ❝♦♥✜r♠ t❤❡ ❋❡❞❡r✲❱❛r❞✐ ❝♦♥❥❡❝t✉r❡ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✸✳ ❚❤❡ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✐♥ ❣❡♥❡r❛❧ ❢♦r ✐♥✜♥✐t❡ str✉❝t✉r❡s✱ ❜✉t ✐t ❞♦❡s ✐❢ t❤❡ str✉❝t✉r❡ ❤❛s ❛ str♦♥❣ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ ♣r♦♣❡rt②✿ ✲❝❛t❡❣♦r✐❝✐t②✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✷ ✴ ✷✹

❈♦♥❥❡❝t✉r❡

❊✈❡r② s❡t ❈❙P ✐s ❡✐t❤❡r ✐♥ P✱ ♦r ◆P✲❝♦♠♣❧❡t❡✳

❯♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤

❋♦r ✜♥✐t❡ str✉❝t✉r❡s✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙Ps ❝❛♥ ❜❡ ❛♥❛❧②s❡❞ ✉s✐♥❣ ❛ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤✳ ❚❤✐s ❛♣♣r♦❛❝❤ ✇❛s ✉s❡❞ ❜② ❇✉❧❛t♦✈✬✵✸ t♦ ❝♦♥✜r♠ t❤❡ ❋❡❞❡r✲❱❛r❞✐ ❝♦♥❥❡❝t✉r❡ ❢♦r ❞♦♠❛✐♥s ♦❢ s✐③❡ ✸✳ ❚❤❡ ✉♥✐✈❡rs❛❧ ❛❧❣❡❜r❛✐❝ ❛♣♣r♦❛❝❤ ❞♦❡s ♥♦t ✇♦r❦ ✐♥ ❣❡♥❡r❛❧ ❢♦r ✐♥✜♥✐t❡ str✉❝t✉r❡s✱ ❜✉t ✐t ❞♦❡s ✐❢ t❤❡ str✉❝t✉r❡ ❤❛s ❛ str♦♥❣ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ ♣r♦♣❡rt②✿ ω✲❝❛t❡❣♦r✐❝✐t②✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✷ ✴ ✷✹

ω✲❝❛t❡❣♦r✐❝✐t②

❆ ✜rst✲♦r❞❡r t❤❡♦r② ✐s ω✲❝❛t❡❣♦r✐❝❛❧ ✐❢ ❛❧❧ ✐ts ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ♠♦❞❡❧s ❛r❡ ✐s♦♠♦r♣❤✐❝✳ ❆ str✉❝t✉r❡ ✐s ω✲❝❛t❡❣♦r✐❝❛❧ ✇❤❡♥❡✈❡r ✐ts ✜rst✲♦r❞❡r t❤❡♦r② ✐s✳ ❋❛❝t✿ ❆ r❡❞✉❝t ♦❢ ❛♥ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡ ✐s ω✲❝❛t❡❣♦r✐❝❛❧✳

❊①❛♠♣❧❡s

❆♥② ✜♥✐t❡ str✉❝t✉r❡ ✐s ✲❝❛t❡❣♦r✐❝❛❧✳ ❚❤❡ t❤❡♦r② ♦❢ ❡✈❡r② ❞❡♥s❡ ❧✐♥❡❛r ♦r❞❡r ✐s ✲❝❛t❡❣♦r✐❝❛❧✳ ❍❡♥❝❡✱ ❛♥❞ ❛r❡ ✲❝❛t❡❣♦r✐❝❛❧✳

✸

✐s ✲❝❛t❡❣♦r✐❝❛❧ s✐♥❝❡ ✐t ✐s ❛ r❡❞✉❝t ♦❢ ✳ ❚❤❡ t❤❡♦r② ♦❢ ❛♥② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ ✐s ✲❝❛t❡❣♦r✐❝❛❧✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✸ ✴ ✷✹

ω✲❝❛t❡❣♦r✐❝✐t②

❆ ✜rst✲♦r❞❡r t❤❡♦r② ✐s ω✲❝❛t❡❣♦r✐❝❛❧ ✐❢ ❛❧❧ ✐ts ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ♠♦❞❡❧s ❛r❡ ✐s♦♠♦r♣❤✐❝✳ ❆ str✉❝t✉r❡ ✐s ω✲❝❛t❡❣♦r✐❝❛❧ ✇❤❡♥❡✈❡r ✐ts ✜rst✲♦r❞❡r t❤❡♦r② ✐s✳ ❋❛❝t✿ ❆ r❡❞✉❝t ♦❢ ❛♥ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡ ✐s ω✲❝❛t❡❣♦r✐❝❛❧✳

❊①❛♠♣❧❡s

❆♥② ✜♥✐t❡ str✉❝t✉r❡ ✐s ω✲❝❛t❡❣♦r✐❝❛❧✳ ❚❤❡ t❤❡♦r② ♦❢ ❡✈❡r② ❞❡♥s❡ ❧✐♥❡❛r ♦r❞❡r ✐s ω✲❝❛t❡❣♦r✐❝❛❧✳ ❍❡♥❝❡✱ (Q, <) ❛♥❞ (R, <) ❛r❡ ω✲❝❛t❡❣♦r✐❝❛❧✳ (Q, {(x, y, z) ∈ Q✸ | (x < y < z) ∨ (z < y < x)}) ✐s ω✲❝❛t❡❣♦r✐❝❛❧ s✐♥❝❡ ✐t ✐s ❛ r❡❞✉❝t ♦❢ (Q, <)✳ ❚❤❡ t❤❡♦r② ♦❢ ❛♥② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r ❛ ✜♥✐t❡ ✜❡❧❞ ✐s ω✲❝❛t❡❣♦r✐❝❛❧✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✸ ✴ ✷✹

■♠♣♦rt❛♥t ❞❡✜♥✐t✐♦♥s

❆ m✲❛r② ♦♣❡r❛t✐♦♥ f ♣r❡s❡r✈❡s ❛ n✲❛r② r❡❧❛t✐♦♥ R ✐❢ ❢♦r ❛❧❧ n✲t✉♣❧❡s x✶, · · · , xm ✐♥ R✱ t❤❡ n✲t✉♣❧❡ (f (x✶,i, · · · , xm,i))✶≤i≤n ✐s ❛❣❛✐♥ ✐♥ R✳ f ✐s ❝❛❧❧❡❞ ❛ ♣♦❧②♠♦r♣❤✐s♠ ♦❢ ❛ r❡❧❛t✐♦♥❛❧ str✉❝t✉r❡ Γ ✐❢ ✐t ♣r❡s❡r✈❡s ❡✈❡r② r❡❧❛t✐♦♥ ♦❢ Γ✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✹ ✴ ✷✹

❯♥✐✈❡rs❛❧ ❆❧❣❡❜r❛✐❝ ❆♣♣r♦❛❝❤ ❛♥❞ ω✲❝❛t❡❣♦r✐❝✐t②

❲❡ ❞❡♥♦t❡ ❜②✿ ✲ P♦❧(Γ) t❤❡ s❡t ♦❢ ❛❧❧ ♣♦❧②♠♦r♣❤✐s♠s ♦❢ Γ✳ ✲ ■♥✈(F) t❤❡ s❡t ♦❢ ❛❧❧ r❡❧❛t✐♦♥s ♣r❡s❡r✈❡❞ ❜② ❛ s❡t F ♦❢ ♦♣❡r❛t✐♦♥s✳ ✲ Γ♣♣ t❤❡ s❡t ♦❢ ❛❧❧ r❡❧❛t✐♦♥s ✇❤✐❝❤ ❛r❡ ♣♣✲❞❡✜♥❛❜❧❡ ♦✈❡r Γ✳

❚❤❡♦r❡♠ ✲ ●❡✐❣❡r ✬✻✽ ✫ ❇♦❞✐rs❦②✲◆❡s❡tr✐❧ ✬✵✸

❋♦r ❡✈❡r② ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ω✲❝❛t❡❣♦r✐❝❛❧ ♦r ✜♥✐t❡ str✉❝t✉r❡ Γ✿ ■♥✈(P♦❧(Γ)) = Γ♣♣

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✺ ✴ ✷✹

▲♦♦❦✐♥❣ ❢♦r ❜❡tt❡r s❡tt✐♥❣s

❋❛❝t✿ ■♥ ❣❡♥❡r❛❧✱ ❛ r❡❞✉❝t ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ B ✐s ♥♦t ω✲❝❛t❡❣♦r✐❝❛❧✳ ◗✉❡st✐♦♥✿ ❈❛♥ ✇❡ ❢♦r♠✉❧❛t❡ ❡✈❡r② s❡t ❈❙P ❛s ❛ ❈❙P ❢♦r ❛♥ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡❄ ❆♥s✇❡r✿ ❨❡s✦ ❉❡✜♥✐t✐♦♥✿ ❆♥ ❡❧❡♠❡♥t ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ❛♥ ❛t♦♠ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ str✐❝t❧② s♠❛❧❧❡r ❡❧❡♠❡♥t t❤❛♥ ♦t❤❡r t❤❛♥ ✵✳ ❊①❛♠♣❧❡✿ t❤❡ s✐♥❣❧❡t♦♥s ❛r❡ t❤❡ ❛t♦♠s ♦❢ ✳

❉❡✜♥✐t✐♦♥ ✲ ❚❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❝♦✉♥t❛❜❧❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✳ ❲❡ ❞❡♥♦t❡ ✐t ❜② ✵ ✶ ✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✻ ✴ ✷✹

▲♦♦❦✐♥❣ ❢♦r ❜❡tt❡r s❡tt✐♥❣s

❋❛❝t✿ ■♥ ❣❡♥❡r❛❧✱ ❛ r❡❞✉❝t ♦❢ t❤❡ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ B ✐s ♥♦t ω✲❝❛t❡❣♦r✐❝❛❧✳ ◗✉❡st✐♦♥✿ ❈❛♥ ✇❡ ❢♦r♠✉❧❛t❡ ❡✈❡r② s❡t ❈❙P ❛s ❛ ❈❙P ❢♦r ❛♥ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡❄ ❆♥s✇❡r✿ ❨❡s✦ ❉❡✜♥✐t✐♦♥✿ ❆♥ ❡❧❡♠❡♥t x ♦❢ ❛ ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ✐s ❝❛❧❧❡❞ ❛♥ ❛t♦♠ ✇❤❡♥ t❤❡r❡ ✐s ♥♦ str✐❝t❧② s♠❛❧❧❡r ❡❧❡♠❡♥t t❤❛♥ x ♦t❤❡r t❤❛♥ ✵✳ ❊①❛♠♣❧❡✿ t❤❡ s✐♥❣❧❡t♦♥s {n} ❛r❡ t❤❡ ❛t♦♠s ♦❢ (P(N), ∩, ∪, ∁, ∅, N)✳

❉❡✜♥✐t✐♦♥ ✲ ❚❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❝♦✉♥t❛❜❧❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✳ ❲❡ ❞❡♥♦t❡ ✐t ❜② A = (A, ⊓, ⊔, c, ✵, ✶)✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✻ ✴ ✷✹

❑❡② ♣r♦♣❡rt②

❋♦r ❡✈❡r② r❡❞✉❝t Γ ♦❢ B✱ t❤❡r❡ ❡①✐sts ❛ r❡❞✉❝t ∆ ♦❢ A ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❈❙P ❛s Γ✳ ❈♦♥✈❡rs❡❧②✱ ❢♦r ❡✈❡r② r❡❞✉❝t ∆ ♦❢ A✱ t❤❡r❡ ❡①✐sts ❛ r❡❞✉❝t Γ ♦❢ B ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❈❙P ❛s ∆✳

❈♦♥s❡q✉❡♥❝❡s

❚♦ s♦❧✈❡ t❤❡ ❝♦♥❥❡❝t✉r❡✱ ✇❡ ❤❛✈❡ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P ❢♦r ❡✈❡r② r❡❞✉❝t ♦❢ ✳ ❙✐♥❝❡ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ❈❙P ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✇❤❡♥ ❛❞❞✐♥❣ ♣♣✲❞❡✜♥❛❜❧❡ r❡❧❛t✐♦♥s✱ ❛♥❞ s✐♥❝❡ ■♥✈ P♦❧

♣♣✱ t❤❡ ❝♦♠♣❧❡①✐t②

♦❢ ❈❙P ✐s ❞❡t❡r♠✐♥❡❞ ❜② P♦❧ ✳ ❈❧❛ss✐❢② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ s❡t ❈❙Ps ✉♥❞❡rst❛♥❞ t❤❡ ♣♦❧②♠♦r♣❤✐s♠s ♦❢ r❡❞✉❝ts ♦❢ ✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✼ ✴ ✷✹

❑❡② ♣r♦♣❡rt②

❋♦r ❡✈❡r② r❡❞✉❝t Γ ♦❢ B✱ t❤❡r❡ ❡①✐sts ❛ r❡❞✉❝t ∆ ♦❢ A ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❈❙P ❛s Γ✳ ❈♦♥✈❡rs❡❧②✱ ❢♦r ❡✈❡r② r❡❞✉❝t ∆ ♦❢ A✱ t❤❡r❡ ❡①✐sts ❛ r❡❞✉❝t Γ ♦❢ B ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❈❙P ❛s ∆✳

❈♦♥s❡q✉❡♥❝❡s

❚♦ s♦❧✈❡ t❤❡ ❝♦♥❥❡❝t✉r❡✱ ✇❡ ❤❛✈❡ t♦ ✉♥❞❡rst❛♥❞ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P(Γ) ❢♦r ❡✈❡r② r❡❞✉❝t Γ ♦❢ A✳ ❙✐♥❝❡ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❛ ❈❙P ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✇❤❡♥ ❛❞❞✐♥❣ ♣♣✲❞❡✜♥❛❜❧❡ r❡❧❛t✐♦♥s✱ ❛♥❞ s✐♥❝❡ ■♥✈(P♦❧(Γ)) = Γ♣♣✱ t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ ❈❙P(Γ) ✐s ❞❡t❡r♠✐♥❡❞ ❜② P♦❧(Γ)✳ ❈❧❛ss✐❢② t❤❡ ❝♦♠♣❧❡①✐t② ♦❢ s❡t ❈❙Ps ⇔ ✉♥❞❡rst❛♥❞ t❤❡ ♣♦❧②♠♦r♣❤✐s♠s ♦❢ r❡❞✉❝ts ♦❢ A✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✼ ✴ ✷✹

❆✉t♦♠♦r♣❤✐s♠s

❙t❛rt✐♥❣ ❜② ❝❧❛ss✐❢②✐♥❣ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s

❆✉t♦♠♦r♣❤✐s♠s ❛r❡ s♣❡❝✐❛❧ ♣♦❧②♠♦r♣❤✐s♠s✳ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ r❡❞✉❝ts ✐s ❛♥ ✐♠♣♦rt❛♥t t♦♣✐❝ ✐♥ t❤❡ t❤❡♦r② ♦❢ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡s✳ ❙✉❝❤ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛s ❜❡❡♥ ❞♦♥❡ ❢♦r (Q, <) ✭❈❛♠❡r♦♥✬✽✾✮✱ t❤❡ r❛♥❞♦♠ ●r❛♣❤ ✭❚❤♦♠❛s✬✾✶✮✱ t❤❡ r❛♥❞♦♠ P♦s❡t ✭P✹✲❙③✬✶✸✮✱ ❡t❝✳✳✳ ❋♦r t❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ✱ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ r❡❞✉❝ts ✐s st✐❧❧ t♦ ❜❡ ❞♦♥❡✳ ❲❡ ❣✐✈❡ s✉❝❤ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ❢♦r ❛♥ ✐♠♣♦rt❛♥t r❡❞✉❝t ♦❢ ✿ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r

✷✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✽ ✴ ✷✹

❆✉t♦♠♦r♣❤✐s♠s

❙t❛rt✐♥❣ ❜② ❝❧❛ss✐❢②✐♥❣ t❤❡ ❛✉t♦♠♦r♣❤✐s♠s

❆✉t♦♠♦r♣❤✐s♠s ❛r❡ s♣❡❝✐❛❧ ♣♦❧②♠♦r♣❤✐s♠s✳ ❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ r❡❞✉❝ts ✐s ❛♥ ✐♠♣♦rt❛♥t t♦♣✐❝ ✐♥ t❤❡ t❤❡♦r② ♦❢ ω✲❝❛t❡❣♦r✐❝❛❧ str✉❝t✉r❡s✳ ❙✉❝❤ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ❤❛s ❜❡❡♥ ❞♦♥❡ ❢♦r (Q, <) ✭❈❛♠❡r♦♥✬✽✾✮✱ t❤❡ r❛♥❞♦♠ ●r❛♣❤ ✭❚❤♦♠❛s✬✾✶✮✱ t❤❡ r❛♥❞♦♠ P♦s❡t ✭P✹✲❙③✬✶✸✮✱ ❡t❝✳✳✳ ❋♦r t❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ A✱ t❤❡ ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ r❡❞✉❝ts ✐s st✐❧❧ t♦ ❜❡ ❞♦♥❡✳ ❲❡ ❣✐✈❡ s✉❝❤ ❛ ❝❧❛ss✐✜❝❛t✐♦♥ ❢♦r ❛♥ ✐♠♣♦rt❛♥t r❡❞✉❝t ♦❢ A✿ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F✷✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✽ ✴ ✷✹

❚❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F✷

❉❡✜♥✐t✐♦♥

❚❤❡r❡ ✐s ❛ ✉♥✐q✉❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F✷✱ ✉♣ t♦ ✐s♦♠♦r♣❤✐s♠✳ ■t ✐s ✐s♦♠♦r♣❤✐❝ t♦ t❤❡ r❡❞✉❝t (A, +) ♦❢ A✱ ✇❤❡r❡ + ❝♦rr❡s♣♦♥❞s t♦ t❤❡ s②♠♠❡tr✐❝ ❞✐✛❡r❡♥❝❡ ♦❢ s❡ts✳ ❋♦r♠❛❧❧②✿ +(x, y, z) ✐❢ ❛♥❞ ♦♥❧② ✐❢ (x ⊔ y) ⊓ (c(x) ⊔ c(y)) = z✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✶✾ ✴ ✷✹

P❛rt✐❛❧ ❝❧❛ss✐✜❝❛t✐♦♥

❚❤❡♦r❡♠ ✲ ❇♦❞✐rs❦②✲❇✳✬✶✸

▲❡t Γ ❜❡ ❛ r❡❞✉❝t ♦❢ (A, +)✱ t❤❡♥✿ ✲ ❡✐t❤❡r ❆✉t(Γ) = ❆✉t(A, +) ✲ ❡✐t❤❡r ❆✉t(Γ) = ❆✉t(A, ❆✛) ✲ ❡✐t❤❡r ❆✉t(Γ) = ❆✉t(A, ✵) ✲ ♦r ❆✉t(Γ) = ❆✉t(A, =) ✇❤❡r❡ ❆✛ =

• (x, y, z, t) ∈ A✹ | x + y = z + t
• ✳

❊①❛♠♣❧❡✿ ❝♦♥s✐❞❡r t❤❡ r❡❧❛t✐♦♥ ❊q✹ ❝♦♥s✐st✐♥❣ ♦❢ ❛❧❧ t✉♣❧❡s (x, y, z, t) s✉❝❤ t❤❛t x + y = z + t ❛♥❞ (x, y, z), (x, z, t), (y, z, t) ❛r❡ ❛ ❢r❡❡ ❢❛♠✐❧✐❡s ♦❢ (A, +)✳ ❚❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣ ♦❢ (A, ❊q✹) ✐s ❆✉t(A, +)✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✵ ✴ ✷✹

❆✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ t❤❡ r❡❞✉❝ts ♦❢ (A, +)

❲❡ ❜❡❧✐❡✈❡ t❤❛t t❤❡ ♣r♦♦❢ t❡❝❤♥✐q✉❡s ❢♦r ♦✉r r❡s✉❧t ❛r❡ ✉s❡❢✉❧ ❛❧s♦ t♦ st✉❞② ♣♦❧②♠♦r♣❤✐s♠s ♦❢ r❡❞✉❝ts ♦❢ (A, +)✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✶ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

• r❛❤❛♠✲▲❡❡❜✲❘♦t❤s❝❤✐❧❞✬✼✷✿ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r

F✷ ❤❛s ❛ str♦♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt② ❝❛❧❧❡❞ ❘❛♠s❡② ♣r♦♣❡rt②✳ ▼❛✐♥ ✐❞❡❛✿ ✇❡ ✉s❡ ❘❛♠s❡② t❤❡♦r② t♦ ♣r♦✈❡ t❤❛t ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ❝❛♥♦♥✐❝❛❧ ♦♥ ❧❛r❣❡ s✉❜s❡ts ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❆ ❢✉♥❝t✐♦♥ ✐s ❝❛❧❧❡❞ ❝❛♥♦♥✐❝❛❧ ✐❢ ❢♦r ❛❧❧

✶

✱ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢

✶

♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢

✶

✳ ❊①❛♠♣❧❡✿ ❧❡t ❜❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ ♦❢ ❛♥❞ ❜❡ ❛ ❜❛s✐s ♦❢ ✳ ❚❤❡ ♠❛♣ ❣❡♥ s✉❝❤ t❤❛t ❣❡♥ ✐s ❝❛♥♦♥✐❝❛❧✳ ❲❡ ✇r✐t❡ ✐❢ ❢♦r ❛❧❧

✶

✱ t❤❡ t②♣❡ ♦❢

✶

❛♥❞ t❤❡ t②♣❡ ♦❢

✶

❛r❡ t❤❡ s❛♠❡✳ ◆♦t❡ t❤❛t ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❲❡ ♣r♦✈❡ t❤❛t ❤❛s ♥✐♥❡ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ♠♦❞✉❧♦ ✳ ❲❡ ❝❧✐♠❜ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ❧❛tt✐❝❡ st❡♣ ❜② st❡♣✱ ♣r♦❞✉❝✐♥❣ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♣❡r❢♦r♠✐♥❣ ❛t ❡❛❝❤ st❡♣ ❛ ❝❛s❡ ❞✐st✐♥❝t✐♦♥✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✷ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

• r❛❤❛♠✲▲❡❡❜✲❘♦t❤s❝❤✐❧❞✬✼✷✿ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r

F✷ ❤❛s ❛ str♦♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt② ❝❛❧❧❡❞ ❘❛♠s❡② ♣r♦♣❡rt②✳ ▼❛✐♥ ✐❞❡❛✿ ✇❡ ✉s❡ ❘❛♠s❡② t❤❡♦r② t♦ ♣r♦✈❡ t❤❛t ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ❝❛♥♦♥✐❝❛❧ ♦♥ ❧❛r❣❡ s✉❜s❡ts ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❆ ❢✉♥❝t✐♦♥ f : A → A ✐s ❝❛❧❧❡❞ ❝❛♥♦♥✐❝❛❧ ✐❢ ❢♦r ❛❧❧ a✶, . . . , an ∈ A✱ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢ (f (a✶), . . . , f (an)) ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢ (a✶, . . . , an)✳ ❊①❛♠♣❧❡✿ ❧❡t (ai)i∈N ❜❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ ♦❢ A ❛♥❞ (bi)i∈N ❜❡ ❛ ❜❛s✐s ♦❢ (A, +)✳ ❚❤❡ ♠❛♣ ❣❡♥: A → A s✉❝❤ t❤❛t ❣❡♥(ai) = bi ✐s ❝❛♥♦♥✐❝❛❧✳ ❲❡ ✇r✐t❡ ✐❢ ❢♦r ❛❧❧

✶

✱ t❤❡ t②♣❡ ♦❢

✶

❛♥❞ t❤❡ t②♣❡ ♦❢

✶

❛r❡ t❤❡ s❛♠❡✳ ◆♦t❡ t❤❛t ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❲❡ ♣r♦✈❡ t❤❛t ❤❛s ♥✐♥❡ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ♠♦❞✉❧♦ ✳ ❲❡ ❝❧✐♠❜ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ❧❛tt✐❝❡ st❡♣ ❜② st❡♣✱ ♣r♦❞✉❝✐♥❣ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♣❡r❢♦r♠✐♥❣ ❛t ❡❛❝❤ st❡♣ ❛ ❝❛s❡ ❞✐st✐♥❝t✐♦♥✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✷ ✴ ✷✹

❙❦❡t❝❤ ♦❢ ♣r♦♦❢

• r❛❤❛♠✲▲❡❡❜✲❘♦t❤s❝❤✐❧❞✬✼✷✿ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r

F✷ ❤❛s ❛ str♦♥❣ ❝♦♠❜✐♥❛t♦r✐❛❧ ♣r♦♣❡rt② ❝❛❧❧❡❞ ❘❛♠s❡② ♣r♦♣❡rt②✳ ▼❛✐♥ ✐❞❡❛✿ ✇❡ ✉s❡ ❘❛♠s❡② t❤❡♦r② t♦ ♣r♦✈❡ t❤❛t ❛✉t♦♠♦r♣❤✐s♠s ❛r❡ ❝❛♥♦♥✐❝❛❧ ♦♥ ❧❛r❣❡ s✉❜s❡ts ♦❢ t❤❡ ❞♦♠❛✐♥✳ ❆ ❢✉♥❝t✐♦♥ f : A → A ✐s ❝❛❧❧❡❞ ❝❛♥♦♥✐❝❛❧ ✐❢ ❢♦r ❛❧❧ a✶, . . . , an ∈ A✱ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢ (f (a✶), . . . , f (an)) ♦♥❧② ❞❡♣❡♥❞s ♦♥ t❤❡ ♠♦❞❡❧✲t❤❡♦r❡t✐❝ t②♣❡ ♦❢ (a✶, . . . , an)✳ ❊①❛♠♣❧❡✿ ❧❡t (ai)i∈N ❜❡ ❛♥ ❡♥✉♠❡r❛t✐♦♥ ♦❢ A ❛♥❞ (bi)i∈N ❜❡ ❛ ❜❛s✐s ♦❢ (A, +)✳ ❚❤❡ ♠❛♣ ❣❡♥: A → A s✉❝❤ t❤❛t ❣❡♥(ai) = bi ✐s ❝❛♥♦♥✐❝❛❧✳ ❲❡ ✇r✐t❡ f ∼ g ✐❢ ❢♦r ❛❧❧ a✶, . . . , an ∈ A✱ t❤❡ t②♣❡ ♦❢ (f (a✶), . . . , f (an)) ❛♥❞ t❤❡ t②♣❡ ♦❢ (g(a✶), . . . , g(an)) ❛r❡ t❤❡ s❛♠❡✳ ◆♦t❡ t❤❛t ∼ ✐s ❛♥ ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳ ❲❡ ♣r♦✈❡ t❤❛t (A, +) ❤❛s ♥✐♥❡ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ♠♦❞✉❧♦ ∼✳ ❲❡ ❝❧✐♠❜ t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ❧❛tt✐❝❡ st❡♣ ❜② st❡♣✱ ♣r♦❞✉❝✐♥❣ ❝❛♥♦♥✐❝❛❧ ❢✉♥❝t✐♦♥s ❛♥❞ ♣❡r❢♦r♠✐♥❣ ❛t ❡❛❝❤ st❡♣ ❛ ❝❛s❡ ❞✐st✐♥❝t✐♦♥✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✷ ✴ ✷✹

P❡rs♣❡❝t✐✈❡s

❖♣❡♥ ♣r♦❜❧❡♠s✿ ❈❧❛ss✐❢② t❤❡ ♣♦❧②♠♦r♣❤✐s♠s ♦❢ r❡❞✉❝ts ♦❢ t❤❡ ❝♦✉♥t❛❜❧② ✐♥✜♥✐t❡ ✈❡❝t♦r s♣❛❝❡ ♦✈❡r F✷✳ ❈❧❛ss✐❢② t❤❡ ❛✉t♦♠♦r♣❤✐s♠ ❣r♦✉♣s ♦❢ r❡❞✉❝ts ♦❢ t❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛✳ ❋✐♥❛❧❧②✿ ❝❧❛ss✐❢② t❤❡ ♣♦❧②♠♦r♣❤✐s♠s ♦❢ r❡❞✉❝ts ♦❢ t❤❡ ❛t♦♠❧❡ss ❇♦♦❧❡❛♥ ❛❧❣❡❜r❛ ❛♥❞ ♦❜t❛✐♥ ❛ ❝♦♠♣❧❡①✐t② ❝❧❛ss✐✜❝❛t✐♦♥ ♦❢ ❛❧❧ s❡t ❈❙Ps✳

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✸ ✴ ✷✹

❚❤❛♥❦ ②♦✉✦

❋r❛♥ç♦✐s ❇♦ss✐èr❡ ✭▲■❳✮ ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ❙❡t ❈❙Ps ❉❡❝❡♠❜❡r ✶✽✱ ✷✵✶✸ ✷✹ ✴ ✷✹