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  1. ❚❤❡ ❊q✉❛t✐♦♥❛❧ ❚❤❡♦r② ♦❢ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❍❛❥♥❛❧ ❆♥❞ré❦❛ ✶ ✱ ❙③❛❜♦❧❝s ▼✐❦✉❧ás ✷ ✱ ■st✈á♥ ◆é♠❡t✐ ✶ ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✶ ❆❧❢ré❞ ❘é♥②✐ ■♥st✐t✉t❡ ♦❢ ▼❛t❤❡♠❛t✐❝s ❍✉♥❣❛r✐❛♥ ❆❝❛❞❡♠② ♦❢ ❙❝✐❡♥❝❡s ✷ ❉❡♣❛rt♠❡♥t ♦❢ ❈♦♠♣✉t❡r ❙❝✐❡♥❝❡ ❛♥❞ ■♥❢♦r♠❛t✐♦♥ ❙②st❡♠s ❇✐r❦❜❡❝❦✱ ❯♥✐✈❡rs✐t② ♦❢ ▲♦♥❞♦♥ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✶ ✴ ✷✶

  2. ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ❚❤❡ ❝❧❛ss ❑❆ ♦❢ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❣❡❜r❛s ♦❢ t❤❡ s✐♠✐❧❛r✐t② t②♣❡ (+ , ; , ∗ , ✵ , ✶ ) s❛t✐s❢②✐♥❣ ❛ ❝❡rt❛✐♥ ✜♥✐t❡ s❡t ♦❢ q✉❛s✐✲❡q✉❛t✐♦♥s ✭❑♦③❡♥✮✳ ❙t❛♥❞❛r❞ ✐♥t❡r♣r❡t❛t✐♦♥s ♦❢ ❑❆ ❛r❡ ❧❛♥❣✉❛❣❡ ❛❧❣❡❜r❛s✱ ▲❑❆✱ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s ❛♥❞ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡s✳ r❡❧❛t✐♦♥ ❛❧❣❡❜r❛s✱ ❘❑❆✱ ❝♦♥♥❡❝t✐♦♥ ✇✐t❤ ♣r♦❣r❛♠ s❡♠❛♥t✐❝s ❛♥❞ ♣r♦♣♦s✐t✐♦♥❛❧ ❞②♥❛♠✐❝ ❧♦❣✐❝ P❉▲✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✷ ✴ ✷✶

  3. ▲❛♥❣✉❛❣❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ▲❡t Σ ❜❡ ❛ s❡t ✭❛❧♣❤❛❜❡t✮ ❛♥❞ Σ ∗ ❞❡♥♦t❡ t❤❡ ❢r❡❡ ♠♦♥♦✐❞ ♦❢ ❛❧❧ ✜♥✐t❡ ✇♦r❞s ♦✈❡r Σ ✱ ✐♥❝❧✉❞✐♥❣ t❤❡ ❡♠♣t② ✇♦r❞ λ ✳ ❚❤❡ ❝❧❛ss ▲❑❆ ♦❢ ❧❛♥❣✉❛❣❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❝❧❛ss ♦❢ s✉❜❛❧❣❡❜r❛s ♦❢ ❛❧❣❡❜r❛s ♦❢ t❤❡ ❢♦r♠ ( ℘ (Σ ∗ ) , + , ; , ∗ , ✵ , ✶ ) + ✐s s❡t ✉♥✐♦♥✱ ; ✐s ❝♦♠♣❧❡① ❝♦♥❝❛t❡♥❛t✐♦♥ ✭♦❢ ✇♦r❞s✮ ❳ ; ❨ = { ①② : ① ∈ ❳ , ② ∈ ❨ } ∗ ✐s t❤❡ ❑❧❡❡♥❡ st❛r ♦♣❡r❛t✐♦♥ ❳ ∗ = { ① ✵ ① ✶ . . . ① ♥ − ✶ : ♥ ∈ ω, ① ✐ ∈ ❳ ❢♦r ❡❛❝❤ ✐ < ♥ } ✵ ✐s t❤❡ ❡♠♣t② ❧❛♥❣✉❛❣❡ ❛♥❞ ✶ ✐s t❤❡ s✐♥❣❧❡t♦♥ ❧❛♥❣✉❛❣❡ ❝♦♥s✐st✐♥❣ ♦❢ t❤❡ ❡♠♣t② ✇♦r❞ λ ✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✸ ✴ ✷✶

  4. ❘❡❧❛t✐♦♥❛❧ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ❚❤❡ ❝❧❛ss ❘❑❆ ♦❢ r❡❧❛t✐♦♥❛❧ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✐s ❞❡✜♥❡❞ ❛s t❤❡ ❝❧❛ss ♦❢ s✉❜❛❧❣❡❜r❛s ♦❢ ❛❧❣❡❜r❛s ♦❢ t❤❡ ❢♦r♠ ( ℘ ( ❲ ) , + , ; , ∗ , ✵ , ✶ ) ✇❤❡r❡ ❲ = ❯ × ❯ ❢♦r s♦♠❡ s❡t ❯ ✱ + ✐s s❡t ✉♥✐♦♥✱ ; ✐s r❡❧❛t✐♦♥ ❝♦♠♣♦s✐t✐♦♥ ① ; ② = { ( ✉ , ✈ ) ∈ ❲ : ( ✉ , ✇ ) ∈ ① ❛♥❞ ( ✇ , ✈ ) ∈ ② ❢♦r s♦♠❡ ✇ } ∗ ✐s r❡✢❡①✐✈❡✲tr❛♥s✐t✐✈❡ ❝❧♦s✉r❡✱ ✵ ✐s t❤❡ ❡♠♣t②s❡t ❛♥❞ ✶ ✐s t❤❡ ✐❞❡♥t✐t② r❡❧❛t✐♦♥ r❡str✐❝t❡❞ t♦ ❲ ✶ = { ( ✉ , ✈ ) ∈ ❲ : ✉ = ✈ } ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✹ ✴ ✷✶

  5. ❊q✉❛t✐♦♥❛❧ t❤❡♦r✐❡s ♦❢ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ▲❑❆ ⊆ ❘❑❆✱ ✇❤❡♥❝❡ ❊q ( ❘❑❆ ) ⊆ ❊q ( ▲❑❆ ) ✳ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ❢ ❛ss✐❣♥s ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ t♦ ❛ ❧❛♥❣✉❛❣❡ ❳ ♦✈❡r ❛♥ ❛❧♣❤❛❜❡t Σ ✿ ❢ ( ❳ ) = { ( ✇ , ✇① ) : ✇ ∈ Σ ∗ ❛♥❞ ① ∈ ❳ } ❚❤❡ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ r❡s♣❡❝ts t❤❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ♦♣❡r❛t✐♦♥s✿ + , ; , ∗ , ✵ , ✶✳ ❇✉t ❘❑❆ �⊆ ▲❑❆✳ ❚❤❡ ✐❞❡♥t✐t② ✶ = { λ } ✐s ❛♥ ❛t♦♠ ✭♠✐♥✐♠❛❧✱ ♥♦♥✲③❡r♦ ❡❧❡♠❡♥t✮ ✐♥ ❧❛♥❣✉❛❣❡ ❛❧❣❡❜r❛s✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✺ ✴ ✷✶

  6. ❊q✉❛t✐♦♥❛❧ t❤❡♦r✐❡s ♦❢ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ▲❑❆ ⊆ ❘❑❆✱ ✇❤❡♥❝❡ ❊q ( ❘❑❆ ) ⊆ ❊q ( ▲❑❆ ) ✳ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ ❢ ❛ss✐❣♥s ❛ ❜✐♥❛r② r❡❧❛t✐♦♥ t♦ ❛ ❧❛♥❣✉❛❣❡ ❳ ♦✈❡r ❛♥ ❛❧♣❤❛❜❡t Σ ✿ ❢ ( ❳ ) = { ( ✇ , ✇① ) : ✇ ∈ Σ ∗ ❛♥❞ ① ∈ ❳ } ❚❤❡ ❈❛②❧❡② r❡♣r❡s❡♥t❛t✐♦♥ r❡s♣❡❝ts t❤❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ♦♣❡r❛t✐♦♥s✿ + , ; , ∗ , ✵ , ✶✳ ❇✉t ❘❑❆ �⊆ ▲❑❆✳ ❚❤❡ ✐❞❡♥t✐t② ✶ = { λ } ✐s ❛♥ ❛t♦♠ ✭♠✐♥✐♠❛❧✱ ♥♦♥✲③❡r♦ ❡❧❡♠❡♥t✮ ✐♥ ❧❛♥❣✉❛❣❡ ❛❧❣❡❜r❛s✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✺ ✴ ✷✶

  7. ❊q✉❛t✐♦♥❛❧ t❤❡♦r✐❡s ♦❢ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✭❝t❞✳✮ ❙❛♠❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r②✿ ❊q ( ❘❑❆ ) = ❊q ( ▲❑❆ ) ✳ ❚❤❡ ❢r❡❡ ❛❧❣❡❜r❛s ♦❢ ❘❑❆ ❛♥❞ ▲❑❆ ❝♦✐♥❝✐❞❡ ✖ ✐t ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ ❤❡♥❝❡ ❛ ❧❛♥❣✉❛❣❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ✭◆é♠❡t✐✮✳ ❋✉rt❤❡r♠♦r❡✱ ❑♦③❡♥✿ ❊q ( ❑❆ ) = ❊q ( ▲❑❆ )(= ❊q ( ❘❑❆ )) ✳ ❚❤✉s t❤❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ♦❢ ❘❑❆ ❛♥❞ ▲❑❆ ✐s ✜♥✐t❡❧② q✉❛s✐✲❛①✐♦♠❛t✐③❛❜❧❡ ✳ ❇✉t ❘❡❞❦♦✿ ❚❤❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ♦❢ ❧❛♥❣✉❛❣❡ ✭r❡❧❛t✐♦♥❛❧✮ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✐s ♥♦t ✜♥✐t❡❧② ❛①✐♦♠❛t✐③❛❜❧❡✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✻ ✴ ✷✶

  8. ❊q✉❛t✐♦♥❛❧ t❤❡♦r✐❡s ♦❢ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✭❝t❞✳✮ ❙❛♠❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r②✿ ❊q ( ❘❑❆ ) = ❊q ( ▲❑❆ ) ✳ ❚❤❡ ❢r❡❡ ❛❧❣❡❜r❛s ♦❢ ❘❑❆ ❛♥❞ ▲❑❆ ❝♦✐♥❝✐❞❡ ✖ ✐t ✐s t❤❡ ❛❧❣❡❜r❛ ♦❢ r❡❣✉❧❛r ❡①♣r❡ss✐♦♥s✱ ❤❡♥❝❡ ❛ ❧❛♥❣✉❛❣❡ ❑❧❡❡♥❡ ❛❧❣❡❜r❛ ✭◆é♠❡t✐✮✳ ❋✉rt❤❡r♠♦r❡✱ ❑♦③❡♥✿ ❊q ( ❑❆ ) = ❊q ( ▲❑❆ )(= ❊q ( ❘❑❆ )) ✳ ❚❤✉s t❤❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ♦❢ ❘❑❆ ❛♥❞ ▲❑❆ ✐s ✜♥✐t❡❧② q✉❛s✐✲❛①✐♦♠❛t✐③❛❜❧❡ ✳ ❇✉t ❘❡❞❦♦✿ ❚❤❡ ❡q✉❛t✐♦♥❛❧ t❤❡♦r② ♦❢ ❧❛♥❣✉❛❣❡ ✭r❡❧❛t✐♦♥❛❧✮ ❑❧❡❡♥❡ ❛❧❣❡❜r❛s ✐s ♥♦t ✜♥✐t❡❧② ❛①✐♦♠❛t✐③❛❜❧❡✳ ❆♥❞ré❦❛✕▼✐❦✉❧ás✕◆é♠❡t✐ ✭✮ ❑❧❡❡♥❡ ▲❛tt✐❝❡s ❚❆❈▲ ✷✵✶✶✱ ✷✾✴✵✼✴✷✵✶✶ ✻ ✴ ✷✶

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