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■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

◆ór❛ ❙③❛❦á❝s

✭❥♦✐♥t ✇♦r❦ ✇✐t❤ ❘♦❜❡rt ●r❛② ❛♥❞ P❡❞r♦ ❙✐❧✈❛✮

❯♥✐✈❡rs✐t② ♦❢ ❨♦r❦✱ ❯❑

❙❛♥❞●❆▲ ✷✵✶✾

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆ ♠♦♥♦✐❞ M ✐s ❝❛❧❧❡❞ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t m ∈ M ❤❛s ❛ ✉♥✐q✉❡ ✐♥✈❡rs❡ m−✶ s❛t✐s❢②✐♥❣ mm−✶m = m, m−✶mm−✶ = m−✶. ❚❤❡ t②♣✐❝❛❧ ❡①❛♠♣❧❡✿ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ ❛ s❡t ✿ ♣❛rt✐❛❧ ✐♥❥❡❝t✐✈❡ ♠❛♣s ✉♥❞❡r ♣❛rt✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ◆❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✿ ✐✛ t❤❡r❡ ❡①✐sts ❛♥ ✐❞❡♠♣♦t❡♥t ✇✐t❤ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆ ♠♦♥♦✐❞ M ✐s ❝❛❧❧❡❞ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t m ∈ M ❤❛s ❛ ✉♥✐q✉❡ ✐♥✈❡rs❡ m−✶ s❛t✐s❢②✐♥❣ mm−✶m = m, m−✶mm−✶ = m−✶. ❚❤❡ t②♣✐❝❛❧ ❡①❛♠♣❧❡✿ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ ❛ s❡t X✿ X → X ♣❛rt✐❛❧ ✐♥❥❡❝t✐✈❡ ♠❛♣s ✉♥❞❡r ♣❛rt✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ◆❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✿ ✐✛ t❤❡r❡ ❡①✐sts ❛♥ ✐❞❡♠♣♦t❡♥t ✇✐t❤ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s

❉❡✜♥✐t✐♦♥

❆ ♠♦♥♦✐❞ M ✐s ❝❛❧❧❡❞ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐❢ ❡✈❡r② ❡❧❡♠❡♥t m ∈ M ❤❛s ❛ ✉♥✐q✉❡ ✐♥✈❡rs❡ m−✶ s❛t✐s❢②✐♥❣ mm−✶m = m, m−✶mm−✶ = m−✶. ❚❤❡ t②♣✐❝❛❧ ❡①❛♠♣❧❡✿ t❤❡ s②♠♠❡tr✐❝ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♦♥ ❛ s❡t X✿ X → X ♣❛rt✐❛❧ ✐♥❥❡❝t✐✈❡ ♠❛♣s ✉♥❞❡r ♣❛rt✐❛❧ ♠✉❧t✐♣❧✐❝❛t✐♦♥✳ ◆❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✿ a ≤ b ✐✛ t❤❡r❡ ❡①✐sts ❛♥ ✐❞❡♠♣♦t❡♥t e ✇✐t❤ a = be✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = InvA | ui = vi (i ∈ I)✱ ✇❤❡r❡ ui, vi ❛r❡ ✇♦r❞s ✐♥ (A ∪ A−✶)∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✱ ✇❤❡r❡ ui = vi✳ ❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r ✿ ❣✐✈❡♥

✶

✱ ❞♦ ✇❡ ❤❛✈❡ ❄ ❤❛s s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❞❡❝✐❞❡s t❤❡ ✇♦r❞ ♣r♦❜❧❡♠✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤❡ t❛❧❦✿ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ s❛t✐s❢② ❛ ❝❡rt❛✐♥ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt② ❤❛✈❡ s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠✱ ✭❛♥❞ ♦t❤❡r ♥✐❝❡ ❛❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s✮✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = InvA | ui = vi (i ∈ I)✱ ✇❤❡r❡ ui, vi ❛r❡ ✇♦r❞s ✐♥ (A ∪ A−✶)∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✱ ✇❤❡r❡ ui = vi✳ ❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r M✿ ❣✐✈❡♥ u, v ∈ (A ∪ A−✶)∗✱ ❞♦ ✇❡ ❤❛✈❡ u =M v❄ ❤❛s s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❞❡❝✐❞❡s t❤❡ ✇♦r❞ ♣r♦❜❧❡♠✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤❡ t❛❧❦✿ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ s❛t✐s❢② ❛ ❝❡rt❛✐♥ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt② ❤❛✈❡ s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠✱ ✭❛♥❞ ♦t❤❡r ♥✐❝❡ ❛❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s✮✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = InvA | ui = vi (i ∈ I)✱ ✇❤❡r❡ ui, vi ❛r❡ ✇♦r❞s ✐♥ (A ∪ A−✶)∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✱ ✇❤❡r❡ ui = vi✳ ❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r M✿ ❣✐✈❡♥ u, v ∈ (A ∪ A−✶)∗✱ ❞♦ ✇❡ ❤❛✈❡ u =M v❄ M ❤❛s s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❞❡❝✐❞❡s t❤❡ ✇♦r❞ ♣r♦❜❧❡♠✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤❡ t❛❧❦✿ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ s❛t✐s❢② ❛ ❝❡rt❛✐♥ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt② ❤❛✈❡ s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠✱ ✭❛♥❞ ♦t❤❡r ♥✐❝❡ ❛❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s✮✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = InvA | ui = vi (i ∈ I)✱ ✇❤❡r❡ ui, vi ❛r❡ ✇♦r❞s ✐♥ (A ∪ A−✶)∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✱ ✇❤❡r❡ ui = vi✳ ❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r M✿ ❣✐✈❡♥ u, v ∈ (A ∪ A−✶)∗✱ ❞♦ ✇❡ ❤❛✈❡ u =M v❄ M ❤❛s s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠ ✐❢ t❤❡r❡ ❡①✐sts ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❞❡❝✐❞❡s t❤❡ ✇♦r❞ ♣r♦❜❧❡♠✳ ❚❤❡ ♠❛✐♥ r❡s✉❧t ♦❢ t❤❡ t❛❧❦✿ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ s❛t✐s❢② ❛ ❝❡rt❛✐♥ ❣❡♦♠❡tr✐❝ ♣r♦♣❡rt② ❤❛✈❡ s♦❧✈❛❜❧❡ ✇♦r❞ ♣r♦❜❧❡♠✱ ✭❛♥❞ ♦t❤❡r ♥✐❝❡ ❛❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s✮✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

▲❡t M ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✳ ❚❤❡ ❈❛②❧❡② ❣r❛♣❤ Γ(M, A) ♦❢ M ✐s ❛♥ ❡❞❣❡✲❧❛❜❡❧❡❞✱ ❞✐r❡❝t❡❞ ❣r❛♣❤

◮ ✇✐t❤ ✈❡rt❡① s❡t M✱ ◮ ❢♦r ❛♥② m ∈ M✱ ❛♥❞ ❛♥② a ∈ A ∪ A−✶✱ m a

− → ma ✐s ❛♥ ❡❞❣❡✳ ◆♦t❡✿

✶✱ ✶

❛r❡ ♥♦t ❛❧✇❛②s ❧♦♦♣s✱ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ✐s ♥♦t str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

▲❡t M ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✳ ❚❤❡ ❈❛②❧❡② ❣r❛♣❤ Γ(M, A) ♦❢ M ✐s ❛♥ ❡❞❣❡✲❧❛❜❡❧❡❞✱ ❞✐r❡❝t❡❞ ❣r❛♣❤

◮ ✇✐t❤ ✈❡rt❡① s❡t M✱ ◮ ❢♦r ❛♥② m ∈ M✱ ❛♥❞ ❛♥② a ∈ A ∪ A−✶✱ m a

− → ma ✐s ❛♥ ❡❞❣❡✳ ◆♦t❡✿

◮ aa−✶✱ a−✶a ❛r❡ ♥♦t ❛❧✇❛②s ❧♦♦♣s✱

t❤❡ ❈❛②❧❡② ❣r❛♣❤ ✐s ♥♦t str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ❈❛②❧❡② ❣r❛♣❤

▲❡t M ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A✳ ❚❤❡ ❈❛②❧❡② ❣r❛♣❤ Γ(M, A) ♦❢ M ✐s ❛♥ ❡❞❣❡✲❧❛❜❡❧❡❞✱ ❞✐r❡❝t❡❞ ❣r❛♣❤

◮ ✇✐t❤ ✈❡rt❡① s❡t M✱ ◮ ❢♦r ❛♥② m ∈ M✱ ❛♥❞ ❛♥② a ∈ A ∪ A−✶✱ m a

− → ma ✐s ❛♥ ❡❞❣❡✳ ◆♦t❡✿

◮ aa−✶✱ a−✶a ❛r❡ ♥♦t ❛❧✇❛②s ❧♦♦♣s✱ ◮ t❤❡ ❈❛②❧❡② ❣r❛♣❤ ✐s ♥♦t str♦♥❣❧② ❝♦♥♥❡❝t❡❞✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts

❋❛❝t✿ ✐❢ m, ma ❛r❡ ✐♥ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ maa−✶ = m✱ t❤❛t ✐s✱ ✐♥ t❤❡s❡ ❝♦♠♣♦♥❡♥ts✱ ❡❞❣❡s ♦❝❝✉r ✐♥ ✐♥✈❡rs❡ ♣❛✐rs✳

❉❡✜♥✐t✐♦♥

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ ✐s ❝❛❧❧❡❞ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ ♦❢ ✱ ❛♥❞ ✐s ❞❡♥♦t❡❞ ❜② ✳ ❋❛❝t✿

✶ ✐s ❛❧✇❛②s ❛ ✈❡rt❡① ♦❢

✱ ♠♦r❡♦✈❡r✱ ✐t ✐s t❤❡ ✉♥✐q✉❡ ✐❞❡♠♣♦t❡♥t ✈❡rt❡①✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts

❋❛❝t✿ ✐❢ m, ma ❛r❡ ✐♥ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ maa−✶ = m✱ t❤❛t ✐s✱ ✐♥ t❤❡s❡ ❝♦♠♣♦♥❡♥ts✱ ❡❞❣❡s ♦❝❝✉r ✐♥ ✐♥✈❡rs❡ ♣❛✐rs✳

❉❡✜♥✐t✐♦♥

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ m ✐s ❝❛❧❧❡❞ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ ♦❢ m✱ ❛♥❞ ✐s ❞❡♥♦t❡❞ ❜② S(m)✳ ❋❛❝t✿

✶ ✐s ❛❧✇❛②s ❛ ✈❡rt❡① ♦❢

✱ ♠♦r❡♦✈❡r✱ ✐t ✐s t❤❡ ✉♥✐q✉❡ ✐❞❡♠♣♦t❡♥t ✈❡rt❡①✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥ts

❋❛❝t✿ ✐❢ m, ma ❛r❡ ✐♥ t❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t✱ t❤❡♥ maa−✶ = m✱ t❤❛t ✐s✱ ✐♥ t❤❡s❡ ❝♦♠♣♦♥❡♥ts✱ ❡❞❣❡s ♦❝❝✉r ✐♥ ✐♥✈❡rs❡ ♣❛✐rs✳

❉❡✜♥✐t✐♦♥

❚❤❡ str♦♥❣❧② ❝♦♥♥❡❝t❡❞ ❝♦♠♣♦♥❡♥t ♦❢ m ✐s ❝❛❧❧❡❞ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤ ♦❢ m✱ ❛♥❞ ✐s ❞❡♥♦t❡❞ ❜② S(m)✳ ❋❛❝t✿ mm−✶ ✐s ❛❧✇❛②s ❛ ✈❡rt❡① ♦❢ S(m)✱ ♠♦r❡♦✈❡r✱ ✐t ✐s t❤❡ ✉♥✐q✉❡ ✐❞❡♠♣♦t❡♥t ✈❡rt❡①✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t❛

▲❡t M = A ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ ❚❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t♦♥ ♦❢ m✿ SA(m) = (S(m), mm−✶, m)✳

❚❤❡♦r❡♠ ✭❙t❡♣❤❡♥✱ ✶✾✾✵✮

◮ L(SA(m)) = {w ∈ (A ∪ A−✶)∗ : w ≥M m}✱ ◮ ❢♦r u, v ∈ (A ∪ A−✶)∗✱ uM = vM ✐✛ v ∈ L(SA(u)) ❛♥❞

u ∈ L(SA(v))✱

◮ t❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r M ❜♦✐❧s ❞♦✇♥ t♦ ❞❡❝✐❞✐♥❣ t❤❡ ❧❛♥❣✉❛❣❡s

♦❢ t❤❡ ❙❝❤üt③❡♥❜❡r❣❡r ❛✉t♦♠❛t❛

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❈♦♥str✉❝t✐♥❣ SA(w)

▲❡t M = InvA | ui = vi (i ∈ I)✱ ❛♥❞ w ∈ (A ∪ A−✶)∗✳ ❚♦ ❜✉✐❧❞ SA(w)✱ ✶✳ st❛rt ✇✐t❤ t❤❡ ❧✐♥❡❛r ❛✉t♦♠❛t♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ w❀ ✷✳ ❡①♣❛♥❞✿ ✐❢ ♦♥❡ s✐❞❡ ♦❢ ❛ r❡❧❛t✐♦♥ ✐s r❡❛❞❛❜❧❡ ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s ❛♥❞ t❤❡ ♦t❤❡r ✐s ♥♦t✱ ❛❞❞ ✐t t♦ t❤❡ ❣r❛♣❤❀ ✸✳ ❢♦❧❞✿ ✐❢ u

a

− →v✶ ❛♥❞ u

a

− →v✷✱ ♦r v✶

a

− →u ❛♥❞ v✷

a

− →u✱ ✐❞❡♥t✐❢② v✶ ❛♥❞ v✷❀ ✹✳ r❡♣❡❛t ✭✷✮ ❛♥❞ ✭✸✮ ✉♥t✐❧ t❤❡ ❣r❛♣❤ st❛❜✐❧✐③❡s ✭♣♦ss✐❜❧② ❢♦r❡✈❡r✮✳

❚❤❡♦r❡♠ ✭❙t❡♣❤❡♥✮

❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❝♦♥✢✉❡♥t✱ ❛♥❞ t❤❡ ❧✐♠✐t ❛✉t♦♠❛t♦♥ ✐s ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❈♦♥str✉❝t✐♥❣ SA(w)

▲❡t M = InvA | ui = vi (i ∈ I)✱ ❛♥❞ w ∈ (A ∪ A−✶)∗✳ ❚♦ ❜✉✐❧❞ SA(w)✱ ✶✳ st❛rt ✇✐t❤ t❤❡ ❧✐♥❡❛r ❛✉t♦♠❛t♦♥ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ w❀ ✷✳ ❡①♣❛♥❞✿ ✐❢ ♦♥❡ s✐❞❡ ♦❢ ❛ r❡❧❛t✐♦♥ ✐s r❡❛❞❛❜❧❡ ❜❡t✇❡❡♥ t✇♦ ✈❡rt✐❝❡s ❛♥❞ t❤❡ ♦t❤❡r ✐s ♥♦t✱ ❛❞❞ ✐t t♦ t❤❡ ❣r❛♣❤❀ ✸✳ ❢♦❧❞✿ ✐❢ u

a

− →v✶ ❛♥❞ u

a

− →v✷✱ ♦r v✶

a

− →u ❛♥❞ v✷

a

− →u✱ ✐❞❡♥t✐❢② v✶ ❛♥❞ v✷❀ ✹✳ r❡♣❡❛t ✭✷✮ ❛♥❞ ✭✸✮ ✉♥t✐❧ t❤❡ ❣r❛♣❤ st❛❜✐❧✐③❡s ✭♣♦ss✐❜❧② ❢♦r❡✈❡r✮✳

❚❤❡♦r❡♠ ✭❙t❡♣❤❡♥✮

❚❤❡s❡ ♦♣❡r❛t✐♦♥s ❛r❡ ❝♦♥✢✉❡♥t✱ ❛♥❞ t❤❡ ❧✐♠✐t ❛✉t♦♠❛t♦♥ ✐s SA(w)✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

• r❛♣❤s ❛s ♠❡tr✐❝ s♣❛❝❡s

▲❡t Γ ❜❡ ❛ ❣r❛♣❤✱ u, v ∈ V (Γ)✱ d(u, v) := min{n : e✶ . . . en ✐s ❛ ♣❛t❤ ❢r♦♠ u t♦ v}. ❚❤✐s ✐s ❛ ♠❡tr✐❝ ♦♥ t❤❡ ✈❡rt✐❝❡s✳ ■❢ ✐s ❛ ❞✐❣r❛♣❤✱ ✐t ♠❛② ♥♦t ❜❡ s②♠♠❡tr✐❝✱ ❤♦✇❡✈❡r ❢♦r ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s✱ ✐t ✐s✳ ❆ ❣❡♦❞❡s✐❝ ❢r♦♠ t♦ ✿ ❛ ♣❛t❤ ✇✐t❤ ❧❡♥❣t❤ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

• r❛♣❤s ❛s ♠❡tr✐❝ s♣❛❝❡s

▲❡t Γ ❜❡ ❛ ❣r❛♣❤✱ u, v ∈ V (Γ)✱ d(u, v) := min{n : e✶ . . . en ✐s ❛ ♣❛t❤ ❢r♦♠ u t♦ v}. ❚❤✐s ✐s ❛ ♠❡tr✐❝ ♦♥ t❤❡ ✈❡rt✐❝❡s✳ ■❢ Γ ✐s ❛ ❞✐❣r❛♣❤✱ ✐t ♠❛② ♥♦t ❜❡ s②♠♠❡tr✐❝✱ ❤♦✇❡✈❡r ❢♦r ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s✱ ✐t ✐s✳ ❆ ❣❡♦❞❡s✐❝ ❢r♦♠ t♦ ✿ ❛ ♣❛t❤ ✇✐t❤ ❧❡♥❣t❤ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

• r❛♣❤s ❛s ♠❡tr✐❝ s♣❛❝❡s

▲❡t Γ ❜❡ ❛ ❣r❛♣❤✱ u, v ∈ V (Γ)✱ d(u, v) := min{n : e✶ . . . en ✐s ❛ ♣❛t❤ ❢r♦♠ u t♦ v}. ❚❤✐s ✐s ❛ ♠❡tr✐❝ ♦♥ t❤❡ ✈❡rt✐❝❡s✳ ■❢ Γ ✐s ❛ ❞✐❣r❛♣❤✱ ✐t ♠❛② ♥♦t ❜❡ s②♠♠❡tr✐❝✱ ❤♦✇❡✈❡r ❢♦r ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s✱ ✐t ✐s✳ ❆ ❣❡♦❞❡s✐❝ ❢r♦♠ u t♦ v✿ ❛ ♣❛t❤ ✇✐t❤ ❧❡♥❣t❤ d(u, v)✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚r❡❡✲❧✐❦❡ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

❆ ❣r❛♣❤ ✐s tr❡❡✲❧✐❦❡ ✐❢ ✐t ✐s q✉❛s✐✲✐s♦♠❡tr✐❝ t♦ ❛ tr❡❡✳

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐s tr❡❡✲❧✐❦❡ ✐❢ ❛❧❧ ✐ts ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡✲❧✐❦❡ ✭t❤✐s ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠✮✳

❊①❛♠♣❧❡s

✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ ✜♥✐t❡ tr❡❡s✮ ✇❤❡r❡ ❛r❡ ❡q✉❛❧ t♦ ✶ ✐♥ t❤❡ ❢r❡❡ ❣r♦✉♣ ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡s✮ ✈✐rt✉❛❧❧② ❢r❡❡ ❣r♦✉♣s

✶ ✶ ✶ ✶

✶ ✭ ✷✮

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚r❡❡✲❧✐❦❡ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

❆ ❣r❛♣❤ ✐s tr❡❡✲❧✐❦❡ ✐❢ ✐t ✐s q✉❛s✐✲✐s♦♠❡tr✐❝ t♦ ❛ tr❡❡✳

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐s tr❡❡✲❧✐❦❡ ✐❢ ❛❧❧ ✐ts ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡✲❧✐❦❡ ✭t❤✐s ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠✮✳

❊①❛♠♣❧❡s

✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ ✜♥✐t❡ tr❡❡s✮ ✇❤❡r❡ ❛r❡ ❡q✉❛❧ t♦ ✶ ✐♥ t❤❡ ❢r❡❡ ❣r♦✉♣ ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡s✮ ✈✐rt✉❛❧❧② ❢r❡❡ ❣r♦✉♣s

✶ ✶ ✶ ✶

✶ ✭ ✷✮

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚r❡❡✲❧✐❦❡ ❣r❛♣❤s

❉❡✜♥✐t✐♦♥

❆ ❣r❛♣❤ ✐s tr❡❡✲❧✐❦❡ ✐❢ ✐t ✐s q✉❛s✐✲✐s♦♠❡tr✐❝ t♦ ❛ tr❡❡✳

❉❡✜♥✐t✐♦♥

❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ✐s tr❡❡✲❧✐❦❡ ✐❢ ❛❧❧ ✐ts ❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡✲❧✐❦❡ ✭t❤✐s ✐s ✐♥❞❡♣❡♥❞❡♥t ♦❢ t❤❡ ❣❡♥❡r❛t✐♥❣ s②st❡♠✮✳

❊①❛♠♣❧❡s

◮ ✜♥✐t❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ◮ ❢r❡❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ ✜♥✐t❡ tr❡❡s✮ ◮ InvA | ui = vi (i ∈ I) ✇❤❡r❡ ui, vi ❛r❡ ❡q✉❛❧ t♦ ✶ ✐♥ t❤❡ ❢r❡❡

❣r♦✉♣ ✭❙❝❤üt③❡♥❜❡r❣❡r ❣r❛♣❤s ❛r❡ tr❡❡s✮

◮ ✈✐rt✉❛❧❧② ❢r❡❡ ❣r♦✉♣s ◮ Invx✶, . . . , xn, y✶, . . . , yn | [x✶, y✶] · . . . · [xn, yn] = ✶ ✭n ≥ ✷✮

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❘❡❣✉❧❛r ❣❡♦❞❡s✐❝s

▲❡t M = X ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ Geo(w) := s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❣❡♦❞❡s✐❝s ✐♥ S(w) ❢r♦♠ x✵ := ww−✶✳

❚❤❡♦r❡♠

■❢ M ✐s ❛ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ Geo(w) ✐s r❡❣✉❧❛r✳ ❘❡♠❛r❦✿ t❤❡r❡ ❡①✐st ❢✳❣✳ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ ❞♦♥✬t ❤❛✈❡ ❛ r❡❣✉❧❛r s❡t ♦❢ ❣❡♦❞❡s✐❝s✱ ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥♦t ✜♥✐t❡❧② ♣r❡s❡♥t❡❞✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❘❡❣✉❧❛r ❣❡♦❞❡s✐❝s

▲❡t M = X ❜❡ ❛♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞✳ Geo(w) := s❡t ♦❢ ❧❛❜❡❧s ♦❢ ❣❡♦❞❡s✐❝s ✐♥ S(w) ❢r♦♠ x✵ := ww−✶✳

❚❤❡♦r❡♠

■❢ M ✐s ❛ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ t❤❡♥ Geo(w) ✐s r❡❣✉❧❛r✳ ❘❡♠❛r❦✿ t❤❡r❡ ❡①✐st ❢✳❣✳ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✇❤✐❝❤ ❞♦♥✬t ❤❛✈❡ ❛ r❡❣✉❧❛r s❡t ♦❢ ❣❡♦❞❡s✐❝s✱ ❛♥❞ ❤❡♥❝❡ ❛r❡ ♥♦t ✜♥✐t❡❧② ♣r❡s❡♥t❡❞✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s

❚❤❡♦r❡♠

■❢ M ✐s ❛ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ S(w) ✐s tr❡❡✲❧✐❦❡✱ t❤❡♥ L(SA(w)) ✐s ❝♦♥t❡①t✲❢r❡❡✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❝♦♥str✉❝ts t❤❡ ♣✉s❤❞♦✇♥ ❛✉t♦♠❛t❛ ✇✐t❤ ✐♥♣✉t t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ ✳ ❍❡♥❝❡✿

❚❤❡♦r❡♠

❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✐s ✉♥✐❢♦r♠❧② ❞❡❝✐❞❛❜❧❡✱ t❤❛t ✐s✱ t❤❡r❡ ✐s ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✇✐t❤ ✐♥♣✉t ✶

✶

t❤❛t ❤❛❧ts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s tr❡❡✲❧✐❦❡✱ ❛♥❞ t❤❡♥ ❞❡❝✐❞❡s ✐❢ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s

❚❤❡♦r❡♠

■❢ M ✐s ❛ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ S(w) ✐s tr❡❡✲❧✐❦❡✱ t❤❡♥ L(SA(w)) ✐s ❝♦♥t❡①t✲❢r❡❡✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❝♦♥str✉❝ts t❤❡ ♣✉s❤❞♦✇♥ ❛✉t♦♠❛t❛ ✇✐t❤ ✐♥♣✉t t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ w✳ ❍❡♥❝❡✿

❚❤❡♦r❡♠

❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✐s ✉♥✐❢♦r♠❧② ❞❡❝✐❞❛❜❧❡✱ t❤❛t ✐s✱ t❤❡r❡ ✐s ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✇✐t❤ ✐♥♣✉t ✶

✶

t❤❛t ❤❛❧ts ✐❢ ❛♥❞ ♦♥❧② ✐❢ ✐s tr❡❡✲❧✐❦❡✱ ❛♥❞ t❤❡♥ ❞❡❝✐❞❡s ✐❢ ✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❡ ♠❛✐♥ t❤❡♦r❡♠s

❚❤❡♦r❡♠

■❢ M ✐s ❛ ✜♥✐t❡❧② ♣r❡s❡♥t❡❞ ✐♥✈❡rs❡ ♠♦♥♦✐❞✱ ❛♥❞ S(w) ✐s tr❡❡✲❧✐❦❡✱ t❤❡♥ L(SA(w)) ✐s ❝♦♥t❡①t✲❢r❡❡✳ ▼♦r❡♦✈❡r✱ t❤❡r❡ ✐s ❛♥ ❛❧❣♦r✐t❤♠ t❤❛t ❝♦♥str✉❝ts t❤❡ ♣✉s❤❞♦✇♥ ❛✉t♦♠❛t❛ ✇✐t❤ ✐♥♣✉t t❤❡ ♣r❡s❡♥t❛t✐♦♥ ❛♥❞ w✳ ❍❡♥❝❡✿

❚❤❡♦r❡♠

❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ✐s ✉♥✐❢♦r♠❧② ❞❡❝✐❞❛❜❧❡✱ t❤❛t ✐s✱ t❤❡r❡ ✐s ❛ ❚✉r✐♥❣ ♠❛❝❤✐♥❡ ✇✐t❤ ✐♥♣✉t

◮ M = InvA | ui = vi (✶ ≤ i ≤ n) ◮ u, v ∈ (A ∪ A−✶)∗

t❤❛t ❤❛❧ts ✐❢ ❛♥❞ ♦♥❧② ✐❢ M ✐s tr❡❡✲❧✐❦❡✱ ❛♥❞ t❤❡♥ ❞❡❝✐❞❡s ✐❢ u =M v✳

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

❚❤❛♥❦ ②♦✉ ❢♦r ②♦✉r ❛tt❡♥t✐♦♥✦

◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s