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  1. ■♥✈❡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

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

  3. ◆❛t✉r❛❧ ♣❛rt✐❛❧ ♦r❞❡r✿ ✐✛ t❤❡r❡ ❡①✐sts ❛♥ ✐❞❡♠♣♦t❡♥t ✇✐t❤ ✳ ■♥✈❡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✐♦♥✳ ◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

  4. ■♥✈❡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

  5. ✶ ❚❤❡ ✇♦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✮✳ ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s ❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = Inv � A | u i = v i ( i ∈ I ) � ✱ ✇❤❡r❡ u i , v i ❛r❡ ✇♦r❞s ✐♥ ( A ∪ A − ✶ ) ∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A ✱ ✇❤❡r❡ u i = v i ✳ ◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

  6. ❤❛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✮✳ ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s ❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = Inv � A | u i = v i ( i ∈ I ) � ✱ ✇❤❡r❡ u i , v i ❛r❡ ✇♦r❞s ✐♥ ( A ∪ A − ✶ ) ∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A ✱ ✇❤❡r❡ u i = v i ✳ ❚❤❡ ✇♦r❞ ♣r♦❜❧❡♠ ❢♦r M ✿ ❣✐✈❡♥ u , v ∈ ( A ∪ A − ✶ ) ∗ ✱ ❞♦ ✇❡ ❤❛✈❡ u = M v ❄ ◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

  7. ❚❤❡ ♠❛✐♥ 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✮✳ ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s ❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = Inv � A | u i = v i ( i ∈ I ) � ✱ ✇❤❡r❡ u i , v i ❛r❡ ✇♦r❞s ✐♥ ( A ∪ A − ✶ ) ∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A ✱ ✇❤❡r❡ u i = v i ✳ ❚❤❡ ✇♦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♦❜❧❡♠✳ ◆ór❛ ❙③❛❦á❝s ❆❧❣♦r✐t❤♠✐❝ ♣r♦♣❡rt✐❡s ♦❢ tr❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s

  8. ■♥✈❡rs❡ ♠♦♥♦✐❞s ❛♥❞ ❧❛♥❣✉❛❣❡s ❚r❡❡✲❧✐❦❡ ✐♥✈❡rs❡ ♠♦♥♦✐❞s ■♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥s ❆♥ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ♣r❡s❡♥t❛t✐♦♥✿ M = Inv � A | u i = v i ( i ∈ I ) � ✱ ✇❤❡r❡ u i , v i ❛r❡ ✇♦r❞s ✐♥ ( A ∪ A − ✶ ) ∗ ✖ t❤❡ ✏♠♦st ❣❡♥❡r❛❧✑ ✐♥✈❡rs❡ ♠♦♥♦✐❞ ❣❡♥❡r❛t❡❞ ❜② A ✱ ✇❤❡r❡ u i = v i ✳ ❚❤❡ ✇♦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

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