Produtos cruzados parciais algébricos e aplicação à Álgebra de Leavitt

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❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛

❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛

P✉r❛ ❡ ❆♣❧✐❝❛❞❛

  

Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s

❛❧❣é❜r✐❝♦s ❡ ❛♣❧✐❝❛çã♦ à ➪❧❣❡❜r❛

❞❡ ▲❡❛✈✐tt

  

●❛❜r✐❡❧❛ ❙✐❧♠❛✐❛ ❞❛ ❙✐❧✈❛ ❨♦♥❡❞❛

❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❉❛♥✐❡❧ ●♦♥ç❛❧✈❡s

  ❋❧♦r✐❛♥ó♣♦❧✐s

  ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛ ❈✉rs♦ ❞❡ Pós✲●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐❝❛❞❛

Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ❛❧❣é❜r✐❝♦s ❡ ❛♣❧✐❝❛çã♦ à ➪❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt

  ❉✐ss❡rt❛çã♦ ❛♣r❡s❡♥t❛❞❛ ❛♦ ❈✉rs♦ ❞❡ Pós✲ ●r❛❞✉❛çã♦ ❡♠ ▼❛t❡♠át✐❝❛ P✉r❛ ❡ ❆♣❧✐✲

  ❝❛❞❛✱ ❞♦ ❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❋ís✐❝❛s ❡ ▼❛t❡♠át✐❝❛s ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡

  ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ♣❛r❛ ❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✱ ❝♦♠ ➪r❡❛ ❞❡

  ❝♦♥❝❡♥tr❛çã♦ ❡♠ ➪❧❣❡❜r❛✳ ▼❛✐♦ ❞❡ ✷✵✶✺

  ❋❧♦r✐❛♥ó♣♦❧✐s

  

através do Programa de Geração Automática da Biblioteca Universitária da UFSC.

Álgebra de Leavitt / Gabriela Silmaia da Silva Yoneda ;

Produtos cruzados parciais algébricos e aplicação à

Yoneda, Gabriela Silmaia da Silva

Ficha de identificação da obra elaborada pelo autor,

Programa de Pós-Graduação em Matemática Pura e Aplicada.

Catarina, Centro de Ciências Físicas e Matemáticas.

Dissertação (mestrado) - Universidade Federal de Santa

69 p.

orientador, Daniel Gonçalves - Florianópolis, SC, 2015.

e Aplicada. III. Título. Santa Catarina. Programa de Pós-Graduação em Matemática Pura Grupóides. I. Gonçalves, Daniel. II. Universidade Federal de

Produtos cruzados parciais. 4. Álgebra de Leavitt. 5.

1. Matemática Pura e Aplicada. 2. Matemática pura. 3.

Inclui referências

  ❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆❣r❛❞❡ç♦ à ♠✐♥❤❛ ❢❛♠í❧✐❛ ❡ ❛♦s ❛♠✐❣♦s✱ ❡st❡❥❛♠ ❡❧❡s ♣❡rt♦✱ ♥❡♠ tã♦ ♣❡rt♦ ♦✉ ❜❛st❛♥t❡ ❧♦♥❣❡✳ ❊♠ ❡s♣❡❝✐❛❧✱ s❡♠♣r❡ ❡s♣❡❝✐❛❧✱ ❛♦ ▲✉✐③✳

  ❆❣r❛❞❡ç♦ ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r✱ Pr♦❢❡ss♦r ❉❛♥✐❡❧✱ ♣❡❧❛ s✉❣❡stã♦ ❞♦ t❡♠❛ ❡ ♣❡❧❛ ❞❡❞✐❝❛çã♦ ❡ ♣❛❝✐ê♥❝✐❛ ♥♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❡st❡ tr❛❜❛❧❤♦✳ ❆❣r❛❞❡ç♦ ❛♦s ♣r♦❢❡ss♦r❡s ❞❛ ❜❛♥❝❛ ♣♦r t❡r❡♠ ❛❝❡✐t❛❞♦ ♦ ❝♦♥✈✐t❡ ❞❡

  ♣❛rt✐❝✐♣❛r ❡ ♣❡❧❛s ❝♦rr❡çõ❡s ❡ s✉❣❡stõ❡s ❢❡✐t❛s✳ ❆❣r❛❞❡ç♦ à ❊❧✐s❛✱ s❡❝r❡t❛r✐❛ ❞❛ Pós✲●r❛❞✉❛çã♦✱ ♣❡❧♦ tr❛❜❛❧❤♦ ✐♠✲

  ♣❡❝á✈❡❧ ❡ ♣♦r t♦❞❛ ❛ ❛❥✉❞❛ ❝♦♠ ❛ ♣❛rt❡ ❜✉r♦❝rát✐❝❛ s❡♠♣r❡✳ ❆❣r❛❞❡ç♦ à ❈❆P❊❙ ✭❈♦♦r❞❡♥❛çã♦ ❞❡ ❆♣❡r❢❡✐ç♦❛♠❡♥t♦ ❞❡ P❡ss♦❛❧

  ❞❡ ◆í✈❡❧ ❙✉♣❡r✐♦r✮ ♣❡❧❛ ❜♦❧s❛ ❞❡ ❡st✉❞♦s ❢♦r♥❡❝✐❞❛✱ s❡♠ ❛ q✉❛❧ ♥ã♦ s❡r✐❛ ♣♦ssí✈❡❧ ❡s❝r❡✈❡r ❡st❛ ❞✐ss❡rt❛çã♦✳

  ❘❡s✉♠♦

  ❉❛❞♦ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E ♣♦❞❡♠♦s ❝♦♥str✉✐r ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡❧❡ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡✲ r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞❡ E ♦✉ ♣♦r ♠❡✐♦ ❞❡ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞❡ E✳ ❊♠ ❛♠❜♦s ♦s ❝❛s♦s t❡♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❞❡ E ❡ ♦s ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ♠❡♥❝✐✲ ♦♥❛❞♦s✳

  ◆❡st❡ tr❛❜❛❧❤♦ ♠♦str❛♠♦s ❛ ❝♦♥str✉çã♦ ❞❡ss❡s ❞♦✐s ♣r♦❞✉t♦s ❝r✉③❛✲ K (E) ❞♦s ♣❛r❝✐❛✐s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ✐s♦♠♦r✜s♠♦s ❝♦♠ L ✳ ❆❧é♠ ❞✐ss♦✱ ❡st✉❞❛♠♦s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❞❛❞♦s ❞♦✐s ❣r❛❢♦s ❞✐r✐❣✐❞♦s E 1

  ❡ E 2 ✱ ❝♦♥s✐❞❡r❛♥❞♦ s❡✉s ❣r✉♣♦✐❞❡s G 1 ❡ G 2 ✱ t❡♥❤❛♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ s✉❛s á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt✳ P♦r ✜♠✱ ❡st✉❞❛♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ❞❛❞♦ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❛s á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt t❡♥❤❛♠♦s ✉♠❛ r❡❧❛✲ çã♦ ♠❛✐s ❢♦rt❡ ❡♥tr❡ ♦s ❣r✉♣♦✐❞❡s✳

  P❛❧❛✈r❛s✲❝❤❛✈❡✿ ➪❧❣❡❜r❛✳ ●r❛❢♦s ❞✐r✐❣✐❞♦s✳ ➪❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt✳ Pr♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳ ●r✉♣♦✐❞❡✳

  ❆❜str❛❝t

  ●✐✈❡♥ ❛ ❞✐r❡❝t❡❞ ❣r❛♣❤ E✱ ♦♥❡ ❝❛♥ ❞❡✜♥❡ ❛ ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣ r✐♥❣ ❛ss♦❝✐❛t❡❞ t♦ ✐t ❜② ❛ ♣❛rt✐❛❧ ❛❝t✐♦♥ ♦❢ t❤❡ ❢r❡❡ ❣r♦✉♣ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡❞❣❡s ♦❢ E ♦r ❜② ❛ ♣❛rt✐❛❧ ❛❝t✐♦♥ ♦❢ t❤❡ ❣r♦✉♣♦✐❞ ❣❡♥❡r❛t❡❞ ❜② t❤❡ ❡❞❣❡s ♦❢ E✳ ■♥ ❜♦t❤ ❝❛s❡s✱ t❤❡r❡ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡ ▲❡❛✈✐tt ♣❛t❤ K (E) ❛❧❣❡❜r❛ L ❛♥❞ t❤❡ ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣✭❣r♦✉♣♦✐❞✮ r✐♥❣ ♠❡♥t✐♦♥❡❞✳

  ■♥ t❤✐s ✇♦r❦✱ ✇❡ s❤♦✇ ❤♦✇ t❤❡s❡ ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣✭❣r♦✉♣♦✐❞✮ r✐♥❣s ❛r❡ ❝♦♥str✉❝t❡❞ ❛♥❞ ✇❡ ❛❧s♦ s❤♦✇ ❤♦✇ t❤❡r❡ ❝❛♥ ❜❡ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡♠✳ ▼♦r❡♦✈❡r✱ ✇❡ st✉❞② s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥s s♦ t❤❛t ❣✐✈❡♥ t✇♦ ❞✐r❡❝t❡❞ ❣r❛♣❤s E 1 ❛♥❞ E 2 ✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡✐r ❣r♦✉♣♦✐❞s G 1 ❛♥❞ G 2 ✱ t❤❡r❡ ✐s ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ t❤❡✐r ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s✳ ❋✐♥❛❧❧②✱

  ✇❡ st✉❞② ❝♦♥❞✐t✐♦♥s s♦ t❤❛t ❣✐✈❡♥ ❛♥ ✐s♦♠♦r♣❤✐s♠ ❜❡t✇❡❡♥ ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s t❤❡r❡ ✐s ❛ str♦♥❣❡r r❡❧❛t✐♦♥ ❜❡t✇❡❡♥ t❤❡ ❣r♦✉♣♦✐❞s✳

  ❑❡②✇♦r❞s✿ ❆❧❣❡❜r❛✳ ❉✐r❡❝t❡❞ ❣r❛♣❤s✳ ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s✳ P❛r✲ t✐❛❧ s❦❡✇ ❣r♦✉♣ r✐♥❣s✳ ●r♦✉♣♦✐❞✳

  ❙✉♠ár✐♦

  

  ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾

  

  ✶✹ ✶✹

  ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✳ ✳ ✸✽

   ✹✼ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✳ ✳ ✳ ✳ ✺✶

  ■♥tr♦❞✉çã♦

  ❆çõ❡s ♣❛r❝✐❛✐s ❞❡ ❣r✉♣♦s ❛♣❛r❡❝❡r❛♠ ✐♥❞❡♣❡♥❞❡♥t❡♠❡♥t❡ ❡♠ ✈ár✐❛s ár❡❛s ❞❛ ♠❛t❡♠át✐❝❛✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ ♥❛ t❡♦r✐❛ ❞❡ á❧❣❡❜r❛ ❞❡ ♦♣❡r❛❞♦✲ r❡s ❝♦♠♦ ✉♠❛ ❢❡rr❛♠❡♥t❛ ♣♦❞❡r♦s❛ ♣❛r❛ s❡✉ ❡st✉❞♦✳ Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s sã♦ ❣❡♥❡r❛❧✐③❛çõ❡s ♥❛t✉r❛✐s ❞❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♥♦ ❝♦♥t❡①t♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ✭✈❡❥❛ ❬✶❪✱ ♦♥❞❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s sã♦ ✐♥tr♦❞✉③✐✲ ❞♦s ❡ s✉❛ ❛ss♦❝✐❛t✐✈✐❞❛❞❡ ❡st✉❞❛❞❛✮✳ ❯♠❛ ❣r❛♥❞❡ ✈❛♥t❛❣❡♠ ❞❡ ❡st✉❞❛r ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s é ❛ s✉❛ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ❢♦r♥❡❝❡r ✉♠ ♠♦❞♦ ❞❡ ❝♦♥str✉✐r ❛♥é✐s ♥ã♦✲❝♦♠✉t❛t✐✈♦s✳ ❆❧é♠ ❞✐ss♦✱ ❤á ✐♥❞í❝✐♦s ❞❡ q✉❡ ❛ t❡♦r✐❛ ❞❡ ❛♥é✐s ♥ã♦✲❝♦♠✉t❛t✐✈♦s ♣♦❞❡ s❡ ❜❡♥❡✜❝✐❛r ❞❛ t❡♦r✐❛ ❞❡ ♣r♦❞✉✲ t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s✳ ❯♠ ❞❡ss❡s ✐♥❞✐❝❛t✐✈♦s é ❛ r❡❝❡♥t❡ ❞❡s❝r✐çã♦ ❞❛s á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt✱ ✉♠❛ ❝❧❛ss❡ ❞❡ á❧❣❡❜r❛ s♦❜r❡ ❝♦r♣♦s ❝♦♥str✉í❞❛ ❞❡ ❣r❛❢♦s ❞✐r✐❣✐❞♦s✱ ❝♦♠♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳

  ❖ ♣r✐♥❝✐♣❛❧ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ é ♠♦str❛r q✉❡ ❛ á❧❣❡❜r❛ ❞❡ ▲❡✲ K (E) ❛✈✐tt L ❞❡ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E é ✐s♦♠♦r❢❛ ❛♦ ♣r✉❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞❡ E✳

  ❊st❡ tr❛❜❛❧❤♦ ❡stá ❞✐✈✐❞✐❞♦ ❡♠ q✉❛tr♦ ❝❛♣ít✉❧♦s✳ ◆♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ✐♥tr♦❞✉③✐r❡♠♦s ❛s ♥♦çõ❡s ❞❡ ❛çõ❡s ♣❛r❝✐❛✐s ❞❡

  ❣r✉♣♦ ❡ ♣r♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ❛ss♦❝✐❛❞♦s ❛ ❡ss❛s ❛çõ❡s✱ ✉s❛♥❞♦ ❝♦♠♦ r❡❢❡rê♥❝✐❛s ♣r✐♥❝✐♣❛✐s ❬✶❪ ❡ ❬✺❪✳ ❆✐♥❞❛ ♥♦ ♣r✐♠❡✐r♦ ❝❛♣ít✉❧♦✱ ❢❛❧❛✲ r❡♠♦s ❞❡ ❝r✐tér✐♦s ♣❛r❛ q✉❡ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ s❡❥❛ s✐♠♣❧❡s✳ ❉❡ ❬✹❪✱ ♦❜t❡r❡♠♦s ❝r✐tér✐♦s ♣❛r❛ ♦ ❝❛s♦ ❞❡ ♦ ❣r✉♣♦ s❡r ❛❜❡❧✐❛♥♦✳

  ◆♦ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦✱ ♥♦s ❞❡❞✐❝❛r❡♠♦s ❛♦ ❡st✉❞♦ ❞❛s á❧❣❡❜r❛s ❞❡ K (E) ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt✱ L ✱ ❡ à ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ L K (E)

  ❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E s♦❜r❡ ✉♠❛ ❞❡t❡r✲ ♠✐♥❛❞❛ á❧❣❡❜r❛✳

  ◆♦ t❡r❝❡✐r♦ ❝❛♣ít✉❧♦✱ ❡st✉❞❛r❡♠♦s ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡ ❡ ♦ ♣r♦✲ ❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡ss❛ ❛çã♦✱ ❜❛s❡❛❞♦s ❡♠ ❬✽❪ ❡ ❬✾❪✳ ▼♦s✲ tr❛r❡♠♦s ♥❡st❡ ❝❛♣ít✉❧♦ q✉❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ❛ á❧❣❡❜r❛ ❞❡

  ▲❡❛✈✐tt ❞❡ ✉♠ ❣r❛❢♦ E ❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞❡ E✳

  ◆♦ ❝❛♣ít✉❧♦ q✉❛tr♦✱ ❡st✉❞❛r❡♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡✱ ❞❛❞♦s ❞♦✐s ❣r✉♣♦✐❞❡s G 1 ❡ G 2 ❛ss♦❝✐❛❞♦s ❛ ❣r❛❢♦s E 1 ❡ E 2 ✱ r❡s♣❡❝t✐✲ ✈❛♠❡♥t❡✱ t❡♥❤❛♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ♣r♦❞✉t♦s ❝r✉③❛❞♦s ❛ss♦✲ ❝✐❛❞♦s ❛ ❡❧❡s✳ ▼♦str❛r❡♠♦s t❛♠❜é♠ ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ❡♠ r❡❧❛çã♦ às á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt ♣❛r❛ q✉❡ t❡♥❤❛♠♦s ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❣r✉♣♦✐❞❡s q✉❡ ♣r❡s❡r✈❛ ❝❛♠✐♥❤♦s ❡ ❞❛♠♦s ✉♠ ❝♦♥tr❛✲❡①❡♠♣❧♦ ❝❛s♦ ✉♠❛ ❞❛s ❝♦♥❞✐çõ❡s ❞❛ ♣r♦♣♦s✐çã♦ ♥ã♦ s❡❥❛ s❛t✐s❢❡✐t❛✳

  ❈❛♣ít✉❧♦ ✶

Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s

❡ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❛çõ❡s ♣❛r❝✐❛✐s ❞❡ ❣r✉♣♦

  ◆❡st❡ ❝❛♣ít✉❧♦ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✶❪✳ ❊♠ s❡❣✉✐❞❛✱ ❞❛❞❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✱ ❝♦♥s✐❞❡✲ r❛r❡♠♦s ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡ss❛ ❛çã♦ ❡ ❞✐s❝✉t✐r❡♠♦s ❝♦♥❞✐çõ❡s ♣❛r❛ q✉❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ s❡❥❛ ❛ss♦❝✐❛t✐✈♦ ✭❝♦♠ ❜❛s❡ ❡♠ ❬✶❪ ❡ ❬✺❪✮ ❡ ♣❛r❛ q✉❡ ❡❧❡ s❡❥❛ s✐♠♣❧❡s ✭❝♦♠ ❜❛s❡ ❡♠ ❬✸❪ ❡ ❬✹❪✮✳ ❖s r❡s✉❧t❛❞♦s q✉❡ ♥ã♦ sã♦ ❞❡♠♦♥str❛❞♦s ❛q✉✐ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ❡♠ ❬✶❪✱ ❬✹❪ ♦✉ ❬✺❪✳

  ✶✳✶ ❆çõ❡s ♣❛r❝✐❛✐s ❞❡ ✉♠ ❣r✉♣♦ G

  ❉❡✜♥✐çã♦ ✶✳✶✳✶✳ ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ X } , {α } )

  é ✉♠ ♣❛r α = ({D t t ∈G t t ∈G ❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ t ∈ G✱ D t é t : D −1 → D t ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ G ❡ α t s❛t✐s❢❛③✿ e = X

  ✭✐✮ D ✱ ❡♠ q✉❡ e ∈ G é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ G❀ −1 (D t ∩ D −1 ) ⊆ D −1

  ✭✐✐✮ α t s (st) ❀ s ◦ α t (x) = α st (x) (D t ∩ D s ) −1 −1 ✭✐✐✐✮ α ✱ ♣❛r❛ t♦❞♦ x ∈ α t

  ❖❜s❡r✈❛çã♦ ✶✳✶✳✷✳ ◆♦t❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠ (iii) ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s✱ −1 (D t ∩ D −1 ) t (x) t (x) ∈ D t ∩ D −1 s❡ x ∈ α t s ✱ ❡♥tã♦ α ❢❛③ s❡♥t✐❞♦ ❡ α s ✳ s t (x)

  ❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r α ❡♠ α ✳ ❆❧é♠ ❞✐ss♦✱ ♦ ✐t❡♠ (ii) st (D t ∩ D −1 ) −1 ❣❛r❛♥t❡ q✉❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛r α ❡♠ x s❡♠♣r❡ q✉❡ x ∈ α t s ✳

  ❖s ✐t❡♥s (ii) ❡ (iii) ❣❛r❛♥t❡♠ ✉♠❛ ❝❡rt❛ ❝♦♠♣❛t✐❜✐❧✐❞❛❞❡ ❡♥tr❡ ❛s ♦♣❡r❛çõ❡s ❞❡ ❝♦♠♣♦s✐çã♦ ❞♦s ✐s♦♠♦r✜s♠♦s ❡ ❛ ♦♣❡r❛çã♦ ❞♦ ❣r✉♣♦✳ ❉❡✜♥✐çã♦ ✶✳✶✳✸✳ ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ✉♠ ❛♥❡❧ R t } t , {α t } t ) t é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ α = ({D ∈G ∈G ❞♦ ❣r✉♣♦ G t❛❧ q✉❡ D é t : D t → D t −1 ✉♠ ✐❞❡❛❧ ❞❡ R ❡ α é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥❡❧✱ ♣❛r❛ t♦❞♦ t ∈ G

  ✳ Pr♦♣♦s✐çã♦ ✶✳✶✳✹✳ ❆s ❝♦♥❞✐çõ❡s ❞❡ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛✿ e = X e = Id X

  ✭✐✮ D ❡ α ❀ t (D −1 ∩ D s ) = D t ∩ D ts ✭✐✐✮ α t ❀ s (α t (x)) = α st (x) −1 ∩ D −1

  ✭✐✐✐✮ α ✱ ♣❛r❛ t♦❞♦ x ∈ D t (st) ❊①❡♠♣❧♦ ✶✳✶✳✺✳ ❙❡❥❛ Z ♦ ❣r✉♣♦ ❛❞✐t✐✈♦ ❞♦s ✐♥t❡✐r♦s ❡ s❡❥❛ N ♦ ❝♦♥❥✉♥t♦ ❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❉❡✜♥❛✱ ♣❛r❛ ❝❛❞❛ z ∈ Z✱ z = N D = {n ∈ N : n ≥ z}. z ◆♦t❡ q✉❡ D ✱ ❝❛s♦ z ≤ 0✳ ❉❡✜♥❛✱ ♣❛r❛ ❝❛❞❛ z ∈ Z✱

  α z : D → D z −z n 7→ n + z z ✳ ❱❡❥❛ q✉❡ α ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ z ∈ Z ♣♦✐s✱ s❡ z ≤ 0 z (n) = n + z ≥ −z + z = 0

  ❡ n ∈ D −z ✱ ❡♥tã♦ n ≥ −z ❡ α ✳ ▲♦❣♦✱ α z (n) ∈ N = D z = N z (n) = n + z ≥ z z (n) ∈ D z ✳ ❙❡ z > 0 ❡ n ∈ D −z ✱ ❡♥tã♦ α ❡ ❝♦♠ ✐ss♦ α ✳ z z ◦ α (n) = z

  ◆♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ z ∈ Z ❡ ♣❛r❛ t♦❞♦ n ∈ D ✱ α −z ✳ −1 = α

  ▲♦❣♦✱ α z −z ✳ z } z , {α z } z ) ❱❛♠♦s ♠♦str❛r q✉❡ α = ({D é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦

  ∈Z ∈Z

  ❣r✉♣♦ Z s♦❜r❡ ♦ ❝♦♥❥✉♥t♦ N✳ = N

  ✭✐✮ P♦r ❞❡✜♥✐çã♦✱ D ✳ −1 (D z ∩ D ) ⊆ D

  ✭✐✐✮ P❛r❛ ♠♦str❛r q✉❡ α z −w −z−w ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r ❞♦✐s ❝❛s♦s✳ −z−w = N ❈❛s♦ ✶✿ −z − w ≤ 0✳ ◆❡st❡ ❝❛s♦✱ D ❡ ❡♥tã♦ s❡❣✉❡ z ∩ D −w ⊆ D −z−w ❞✐r❡t❛♠❡♥t❡ q✉❡ D ✳ ❈❛s♦ ✷✿ −z − w > 0✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s z < −w✱ ♦ q✉❡ ✐♠♣❧✐❝❛ z ∩ D = D ❡♠ D −w −w ✳ ❉❛í✱ ♣❛r❛ x ∈ D −w ✱ t❡♠♦s x ≥ −w (x) = x − z ≥ −w − z.

  ❡ α −z

  (x) ∈ D ▲♦❣♦✱ α −z −w−z ✳

  (D z ∩ D ) ⊆ D P♦rt❛♥t♦✱ α −z −w −z−w ✳ −1

  (D z ∩ D ) ✭✐✐✐✮ ❙❡❥❛♠ z, w ∈ Z ❡ s❡❥❛ n ∈ α z −w ✳ ❊♥tã♦ α w ◦ α z (n) = α w (z + n) = w + (z + n) = (w + z) + n = α w +z (n). g } g , {α g } g )

  ❊①❡♠♣❧♦ ✶✳✶✳✻✳ ❙❡❥❛ α = ({D ∈G ∈G ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ X ❡ s❡❥❛ K ✉♠ ❝♦r♣♦ q✉❛❧q✉❡r✳ X

  = {f : X → K|f ❙❡❥❛ K é ❢✉♥çã♦ }✳ ❈♦♥s✐❞❡r❡✱ ♣❛r❛ ❝❛❞❛ g ∈ G✱ X −1 E g = {f ∈ K : f | X = 0}. \D g

  P❛r❛ t♦❞❛ f ∈ E g ✱ ♣❛r❛ t♦❞♦ x ∈ X✱ s❡❥❛ ( −1 f (α g (x)) g ✱ s❡ x ∈ D

  β g (f )(x) = g .

  ✱ s❡ x /∈ D g g g g : E −1 → E ❱❡❥❛ q✉❡ β é ✉♠❛ ❢✉♥çã♦ β ✳

  } , {β } ) ❱❛♠♦s ♠♦str❛r q✉❡ β = ({E g g ∈G g g ∈G é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ X G s♦❜r❡ K ✳

  ✭✐✮ ◆♦t❡ q✉❡ X E e = {f ∈ K : f | = 0} X X \D e = {f ∈ K : f | X X } \X=0 = K . g ∩ E h ) −1

  ✭✐✐✮ P❛r❛ q✉❛✐sq✉❡r g, h ∈ G ❡ f ∈ (E ✱ ✈❛♠♦s ♠♦str❛r q✉❡ β g (f ) ∈ E gh gh

  ✳ ❙❡❥❛ x ∈ X \ D ✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s✳ g ❈❛s♦ ✶✿ x /∈ D ✳ g (f )(x) = 0 ❊♥tã♦ β ✱ ♣♦r ❞❡✜♥✐çã♦✳ g ❈❛s♦ ✷✿ x ∈ D ✳ g \ D gh = D g \ (D gh ∩ D g ) ◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡ x ∈ D ✳ ▲♦❣♦✿

  α −1 (x) ∈ α −1 (D g \ (D g ∩ D gh )) = D −1 \ α −1 (D g ∩ D gh ) g g g g −1 (x) ∈ D −1 \ (D −1 ∩ D h ) ⊆ X \ D h = D −1 \ (D −1 ∩ D h ). g g ❖✉ s❡❥❛✱ α g g g ✳ g (f )(x) = f (α −1 (x)) = 0 = 0 ▲♦❣♦✱ β g ✱ ♣♦✐s f| X \D h ✱ ✈✐st♦ q✉❡ f ∈ E h g (f )| = 0 g (E −1 ∩E h ) ⊆ E gh

  ✳ P♦rt❛♥t♦✱ β X \D gh ✳ ■st♦ é✱ β g ✳

  ✭✐✐✐✮ P❛r❛ q✉❛✐sq✉❡r g, h ∈ G, f ∈ E h −1 ∩ E h −1 g −1 ❡ x ∈ X✱ t❡♠♦s β g (β h (f ))(x) = (

  β h (f )(α g −1 (x)) ✱ s❡ x ∈ D g ✱ s❡ x /∈ D g

  = f (α h −1 (α g −1 (x))) ✱ s❡ x ∈ D g , α g −1

  (x) ∈ D h ✱ s❡ x ∈ D g , α g −1 (x) / ∈ D h ✱ s❡ x /∈ D g .

  ◆♦t❡ q✉❡ x ∈ D g ❡ α g −1 (x) ∈ D h ⇔ x ∈ D g ❡ α g −1 (x) ∈ D h ∩ D g −1

  ⇔ x ∈ D g ❡ x = α g (α g −1 (x)) ∈ α g (D h ∩ D g −1 )

  ⇔ x ∈ D g ❡ x ∈ D (g −1 ) −1 ∩ D (g −1 ) −1 (h −1 ) −1 ⇔ x ∈ D g ∩ D gh .

  ▲♦❣♦✱ β g (β h (f ))(x) = ( f (α h −1 (α g −1 (x)))

  ✱ s❡ x ∈ D g ∩ D gh ✱ ❝❛s♦ ❝♦♥trár✐♦

  = f (α h −1 g −1 (x)) ✱ s❡ x ∈ D gh ∩ D g ✱ s❡ x ∈ D gh \ D g ✱ s❡ x /∈ D gh .

  ❱❡❥❛ q✉❡ s❡ x ∈ D gh \ D g = D gh \ (D gh ∩ D g ) ✱ s❡❣✉❡ q✉❡ α h −1 g −1 (x) ∈ α h −1 g −1 (D gh \ (D gh ∩ D g )) = D h −1 g −1 \ α h −1 g −1 (D gh ∩ D g )

  = D h −1 g −1 \ (D h −1 g −1 ∩ D h −1 ) = D h −1 g −1 \ D h −1 ⊆ X \ D h −1

  ❈♦♠♦ f ∈ E h −1 ✱ ♦❜t❡♠♦s f(α h −1 g −1 ) = 0 ✳ ❉✐ss♦ s❡❣✉❡ q✉❡ β g (β h (f ))(x) = f (α h −1 g −1 (x)), s❡ x ∈ D g ∩ D gh f (α h −1 g −1 (x)), s❡ x ∈ D gh \ D g

  0, s❡ x /∈ D gh = ( f (α h −1 g −1 (x)), s❡ x ∈ D gh

  ✱ s❡ x /∈ D gh .

  = β gh (f )(x). ❈♦♠♦ ♦✉tr♦s ❝❛s♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ t❡♠♦s q✉❡ ( −1 f (α (x)) gh (gh) ✱ s❡ x ∈ D

  β g (β h (f ))(x) = gh .

  ✱ s❡ x /∈ D −1 ∩ D −1 −1 P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r g, h ∈ G ❡ f ∈ D h h g ✱ t❡♠♦s β g (β h (f )) = β gh (f ).

  ✶✳✷ Pr♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ t } t , {α t } t )

  ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❝♦r♣♦ K ❡ s❡❥❛ α = ({D ∈G ∈G ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G s♦❜r❡ ❛ á❧❣❡❜r❛ A✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥✲ X a g δ g g ∈ D g

  ❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦♠❛s ❢♦r♠❛✐s ✜♥✐t❛s ❞❛ ❢♦r♠❛ ✱ ❝♦♠ a ✳ g ∈G ❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s ♥♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ❛❝✐♠❛✿ X X X

  • a g δ g b g δ g = (a g + b g )δ g ; ❼ g g g ∈G ∈G ∈G
  • X X a g δ g = (λa g )δ g ❼ λ ✱ ❡♠ q✉❡ λ ∈ K❀ g ∈G g ∈G g δ g ) · (b h δ h ) = α g (α (a g )b h )δ gh . −1 ❼ (a g ❊ ❡ss❛ ♦♣❡r❛çã♦ ❡st❡♥❞❡✲s❡

      ❧✐♥❡❛r♠❡♥t❡ ♣❛r❛ t♦❞♦ ♦ ❝♦♥❥✉♥t♦✳ X a g δ g : a g ∈ D g } ❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❖ ❝♦♥❥✉♥t♦ { ♠✉♥✐❞♦ ❞❛s ♦♣❡✲ g ∈G r❛çõ❡s ❛❝✐♠❛ é ❞❡♥♦♠✐♥❛❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ α ❡ α G

      ❞❡♥♦t❛❞♦ ♣♦r A ⋊ ✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳✷✳ ❱❡❥❛ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s✱ −1 −1 (a g )b h ∈ D −1 ❝♦♠♦ D g é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡ A✱ ❡♥tã♦ α g g ✳ α

      G, a 7→ aδ e ❖❜s❡r✈❛çã♦ ✶✳✷✳✸✳ ❆ ❛♣❧✐❝❛çã♦ ι : A → A ⋊ é ✉♠ ❤♦✲ ♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛ ✐♥❥❡t✐✈♦✱ ❧♦❣♦ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A ❝♦♠ ✉♠❛ α G s✉❜á❧❣❡❜r❛ ❞❡ A ⋊ ✳ A ⋊ α G

      é ✉♠❛ á❧❣❡❜r❛ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ss♦❝✐❛t✐✈❛✳ ❖✉ s❡❥❛✱ ❡♠ g δ g ·b h δ h )·c j δ j = a g δ g ·(b h δ h ·c j δ j ) ❣❡r❛❧✱ ♥ã♦ é ✈❡r❞❛❞❡ q✉❡ (a ✳ ❚♦❞❛✈✐❛✱ ❝♦♠♦ é ♠♦str❛❞♦ ❡♠ ❬✶❪✱ ✈❛❧❡ ♦ r❡s✉❧t❛❞♦ q✉❡ s❡❣✉❡ ❛s ❞❡✜♥✐çõ❡s ❛❜❛✐①♦✳

      ❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❡ s❡❥❛ I ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ 2 A ✳ ❉✐③❡♠♦s q✉❡ I é ✐❞❡♠♣♦t❡♥t❡ s❡ I = I ✱ ✐st♦ é✱ s❡ t♦❞♦ ❡❧❡♠❡♥t♦ ❞❡

      I é ✉♠❛ s♦♠❛ ❞❡ ♣r♦❞✉t♦s ❞❡ ♦✉tr♦s ❡❧❡♠❡♥t♦s ❞❡ I✳

      ❉❡✜♥✐çã♦ ✶✳✷✳✺✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❡ s❡❥❛ I ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A✳ ❉✐③❡♠♦s q✉❡ I é ♥ã♦✲❞❡❣❡♥❡r❛❞♦ s❡ ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ♥ã♦✲♥✉❧♦ a ∈ I ❡①✐st❡ b ∈ I t❛❧ q✉❡ ab 6= 0 ♦✉ ba 6= 0✳ ❈♦r♦❧ár✐♦ ✶✳✷✳✻✳ ❙❡❥❛ α ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G ✉♠ ✉♠❛ t á❧❣❡❜r❛ A t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ t ∈ G✱ t❡♠✲s❡ q✉❡ D é ✐❞❡♠♣♦t❡♥t❡ ♦✉ α G ♥ã♦✲❞❡❣❡♥❡r❛❞♦✳ ❊♥tã♦ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ A⋊ é ❛ss♦❝✐❛t✐✈♦✳ α G

      ❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ♥♦s ❞á ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡ A ⋊ s❡❥❛ ❛ss♦❝✐❛t✐✈♦✱ s❡♥❞♦ q✉❡ t❛❧ ❝♦♥❞✐çã♦ ❞❡♣❡♥❞❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s t α G ✐❞❡❛✐s D ✳ ❊♠ s❡❣✉✐❞❛✱ ♥♦s ♣r❡♦❝✉♣❛♠♦s ❝♦♠ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ A⋊ ✳ t } t , {α t } t )

      ❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ α = ({D ∈G ∈G ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ G ❡♠ ✉♠ ❛♥❡❧ R✱ ❡♠ ❬✹❪✱ ❡st✉❞❛♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssá✲ α G r✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ A ⋊ s❡❥❛ s✐♠♣❧❡s✳

      ❉✉r❛♥t❡ ♦s ♥♦ss♦s ❡st✉❞♦s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❣r✉♣♦ G é ❛❜❡❧✐❛♥♦ t ❡ ❝❛❞❛ D ♣♦ss✉✐ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ✭❞❡✜♥✐r❡♠♦s ♦ q✉❡ ✐ss♦ s✐❣♥✐✜❝❛ ❧♦❣♦ ❡♠ s❡❣✉✐❞❛✮✱ ❝♦♥s❡❣✉✐♠♦s ❡♥❢r❛q✉❡❝❡r ❛ ❝♦♥❞✐çã♦ ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪ α G q✉❡ ❣❛r❛♥t❡ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ A ⋊ ✳

      ❆❣♦r❛ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦s ♣ré✲r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❡♥✉♥❝✐❛r✲ ♠♦s ♦ t❡♦r❡♠❛ ♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✳ ◆♦ q✉❡ s❡❣✉❡✱ G é ✉♠ ❣r✉♣♦ ❛❜❡❧✐✲ ❛♥♦✱ A é ✉♠ ❛♥❡❧ ❡ ❡stã♦ ♦♠✐t✐❞❛s ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s q✉❡ ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✹❪✳ X a t δ t ∈ A ⋊ α G

      ❉❡✜♥✐çã♦ ✶✳✷✳✼✳ ❙❡❥❛ a = ✳ ❉❡✜♥✐♠♦s✿ t ∈G ✭✐✮ ♦ s✉♣♦rt❡ ❞❡ a✱ ❞❡♥♦t❛❞♦ ♣♦r supp(a)✱ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦

      {t ∈ G : a t 6= 0}; g : A ⋊ α G → A ✭✐✐✮ ❛ ♣r♦❥❡çã♦ ❞❡ a ♥❛ ❝♦♦r❞❡♥❛❞❛ g✱ P ✱ ♣♦r X P g ( a t δ t ) = a g . t ∈G

      ❉❡✜♥✐çã♦ ✶✳✷✳✽✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❉✐③❡♠♦s q✉❡ A ♣♦ss✉✐ ✉♥✐❞❛❞❡s , r , . . . , r n } ⊆ A

      ❧♦❝❛✐s s❡ ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦ {r 2 1 2 ❡①✐st❡ e ∈ A t❛❧ = e i = r i = r i e , q✉❡ e ❡ er ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✳

      ❉❡✜♥✐çã♦ ✶✳✷✳✾✳ ❯♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s E ⊆ A é ✉♠ ❝♦♥✲ , . . . , r n }

      ❥✉♥t♦ ❞❡ ✐❞❡♠♣♦t❡♥t❡s t❛✐s q✉❡ ♣❛r❛ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ {r i = r i e = r i 1 ❞❡ A ❡①✐st❡ e ∈ E t❛❧ q✉❡ er ✱ ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✳ Pr♦♣♦s✐çã♦ ✶✳✷✳✶✵✳ ❙❡❥❛ E ⊆ A ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A = {eδ : e ∈ E}

      ✳ ❊♥tã♦ Eδ é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A ⋊ α G

      ✳ 1 n , . . . , a } α G

      ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ {a ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡ A ⋊ ✱ i X i = a δ t

      ❡ ❡s❝r❡✈❡♠♦s a t ✱ ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✳ ❊♥tã♦ ❡①✐st❡ t ∈G i s♦♠❡♥t❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ a t ✬s q✉❡ sã♦ ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✳ t (a ) : a 6= 0} ∪ {a : a 6= 0} −1 i i i i ❙❡❥❛ X = {α t t t t ✳ ❈♦♠♦ X é ✜♥✐t♦ ❡ A

      ♣♦ss✉✐ ✉♥✐❞❛❞❡s ❧♦❝❛✐s✱ s❡❥❛ e ∈ E ✉♥✐❞❛❞❡ ❧♦❝❛❧ ♣❛r❛ X✳ ❉❛í✱ i i i i i i i eδ a δ t = ea δ t = a δ t δ t eδ = α t (α t −1 (a )e)δ t = α t (α t −1 (a ))δ t = a δ t . t t t ❡ a t t t t ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ 2

      (eδ ) = eδ · eδ = α (α (e) · e)δ = α (e · e)δ = α (e)δ = eδ , ❝♦♠♦ q✉❡rí❛♠♦s✳ t } t , {α t } t ) ❉❡✜♥✐çã♦ ✶✳✷✳✶✶✳ ❙❡❥❛ α = ({D ∈G ∈G ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G ❡♠ ✉♠ ❛♥❡❧ A✳ ❉✐③❡♠♦s q✉❡ ✉♠ ✐❞❡❛❧ I E A é ●✲✐♥✈❛r✐❛♥t❡ g (I ∩ D −1 ) ⊆ I ∩ D g s❡ α g ✱ ♣❛r❛ t♦❞♦ g ∈ G✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶✷✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❛♥❡❧ A é ●✲s✐♠♣❧❡s s❡ ♦s ú♥✐❝♦s ✐❞❡❛✐s ●✲✐♥✈❛r✐❛♥t❡s ❞❡ A sã♦ A ❡ {0}✳ ▲❡♠❛ ✶✳✷✳✶✸✳ ❙❡❥❛ E ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A✳ ❙❡❥❛ α t

      ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✭❛❞✐t✐✈♦✮ G t❛❧ q✉❡ D t❡♠ ✉♥✐❞❛❞❡s ❧♦❝❛✐s✱ ♣❛r❛ t♦❞♦ t ∈ G✳ ❙✉♣♦♥❤❛ q✉❡ A é ●✲s✐♠♣❧❡s✳ α G

      ❊♥tã♦✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦ r ∈ A⋊ ✱ ❡ ♣❛r❛ ❝❛❞❛ ✉♥✐❞❛❞❡ ∈ A ⋊ α G =: R

      ❧♦❝❛❧ e ∈ E✱ ❡①✐st❡ r t❛❧ q✉❡✿ ∈ RrR

      ❛✳ r ❀ (r ) = e

      ❜✳ P ❀ ) ≤ # supp(r)

      ❝✳ # supp(r ✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶✹✳ ❙❡❥❛ R ✉♠ ❛♥❡❧✳ ❖ ❝❡♥tr♦ ❞❡ R é ♦ ❝♦♥❥✉♥t♦ C(R) = {a ∈ R : ab = ba,

      ♣❛r❛ t♦❞♦ b ∈ R}✳

      ❖❜s❡r✈❛çã♦ ✶✳✷✳✶✺✳ ◆♦t❡ q✉❡ C(R) 6= ∅✱ ♣♦✐s 0 ∈ C(R)✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ C(R) ❡ r ∈ R t❡♠♦s (a−b)r = ar−br = ra−rb = r(a − b)

      ❡ (ab)r = a(br) = a(rb) = (ar)b = (ra) = r(ab)✳ ❖✉ s❡❥❛✱ C(R) é ✉♠ s✉❜❛♥❡❧ ❝♦♠✉t❛t✐✈♦ ❞❡ R✳ e Seδ

      P❛r❛ A⋊G =: S ❡ E ⊆ A✱ ❞❡♥♦t❛r❡♠♦s ♣♦r C ♦ ❝❡♥tr♦ ❞❡ eδ ✳ α G ❉❡✜♥✐çã♦ ✶✳✷✳✶✻✳ ❙❡❥❛♠ R = A ⋊ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❡ E ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A✳ P❛r❛ ❝❛❞❛ e ∈ E✱ ❞❡✜♥✐♠♦s ♦

      Reδ ❝❡♥tr♦ ❞❡ eδ ❝♦♠♦ C e := {x ∈ eδ Reδ : xy = yx, Reδ }.

      ♣❛r❛ t♦❞♦ y ∈ eδ ▲❡♠❛ ✶✳✷✳✶✼✳ ❈♦♥s✐❞❡r❡ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r ❡ s❡❥❛ e ∈ E α G

      ✳ ❊♥tã♦ t♦❞♦ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A ⋊ t❡♠ ✐♥t❡rs❡çã♦ ♥ã♦✲♥✉❧❛ X

    • e ∩ {eδ b g δ g } ❝♦♠ C ✳ g ∈G\{0}

      P♦r ✜♠✱ ❡♥✉♥❝✐❛♠♦s ♦ t❡♦r❡♠❛✳ ❊♠ ❬✹❪✱ é ♠♦str❛❞♦ q✉❡ ♦s ✐t❡♥s (i)

      ❡ (ii) sã♦ ❡q✉✐✈❛❧❡♥t❡s✳ ❚❡♦r❡♠❛ ✶✳✷✳✶✽✳ ❙❡❥❛ E ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A ❡ t s❡❥❛ α ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ G t❛❧ q✉❡ t♦❞♦ ✐❞❡❛❧ D t❡♠ ✉♥✐❞❛❞❡s ❧♦❝❛✐s✳ ❊♥tã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ α G

      ✭✐✮ A ⋊ é s✐♠♣❧❡s❀ e ✭✐✐✮ A é ●✲s✐♠♣❧❡s ❡ C é ❝♦r♣♦✱ ♣❛r❛ t♦❞♦ e ∈ E❀ e

      ✭✐✐✐✮ A é ●✲s✐♠♣❧❡s ❡ C é ❝♦r♣♦✱ ♣❛r❛ ❛❧❣✉♠ e ∈ E✳ ❉❡♠♦♥str❛çã♦✿ (ii) ⇒ (iii) ➱ ó❜✈✐♦✳

      (iii) ⇒ (i) α G ❙❡❥❛ J ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A ⋊ ✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ e ) \ {0} e = r · r ∈ J −1

      ❡①✐st❡ r ∈ (J ∩ C ✳ ❈♦♠♦ C é ❝♦r♣♦✱ ❡♥tã♦ eδ ✳ α G a 7→ aδ ❈♦♥s✐❞❡r❡ ♦ ♠♦r✜s♠♦ ϕ : A → A ⋊ ✳ ❊♥tã♦ é ❢á❝✐❧ ✈❡r −1

      (J) q✉❡ ϕ é ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A✳ −1 −1 (J) (J) ∩ D

      ❱❛♠♦s ♠♦str❛r q✉❡ ϕ é ●✲✐♥✈❛r✐❛♥t❡✳ ❙❡❥❛ a ∈ ϕ −h ✳ h ❙❡❥❛ e ✉♥✐❞❛❞❡ ♣❛r❛ a ❡♠ D −h ✳ ❉❛í✱

      α (e )δ · aδ · e δ = α (α (α (e )a))δ · e δ h h h h −h h −h h h h h −h = α h (e h a)δ h · e h δ −h = α h (a)δ h · e h δ −h = α h (α (a)e h )δ

      = α h (a)δ . ▲♦❣♦ α h (a)δ ∈ J ❡ ❡♥tã♦ α h (a) ∈ ϕ −1

      (J) ✱ ❞♦♥❞❡ ϕ −1

      (J) é G✲

      ✐♥✈❛r✐❛♥t❡ ❡ ϕ −1 (J) = A

      ✱ ♣♦✐s A é G✲s✐♠♣❧❡s ❡ ϕ −1 (J) 6= 0

      ✳ P♦rt❛♥t♦✱ aδ ∈ J ♣❛r❛ t♦❞♦ a ∈ A✳

      ❆❣♦r❛✱ s❡❥❛ a g ∈ D g q✉❛❧q✉❡r✳ ❙❡❥❛ e g ✉♥✐❞❛❞❡ ♣❛r❛ a g ∈ D g ✳ ❉❛í✱ a g δ · e g δ g = α (α (a g )e g )δ g = α (a g e g )δ g = a g δ g . ▲♦❣♦✱ a g δ g ∈ J, ♣❛r❛ t♦❞♦ a g ∈ D g , ♣❛r❛ t♦❞♦ g ∈ G✳ ❉♦♥❞❡✱

      J = A ⋊ α G ❡✱ ♣♦rt❛♥t♦✱ A ⋊ α G é s✐♠♣❧❡s✳

      ❈❛♣ít✉❧♦ ✷ ❆s á❧❣❡❜r❛s ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt ❝♦♠♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧

      ◆❡st❡ ❝❛♣ít✉❧♦ ♥♦s ❝♦♥❝❡♥tr❛♠♦s ❡♠ ✉♠❛ ❝❧❛ss❡ ❡s♣❡❝í✜❝❛ ❞❡ ♣r♦✲ ❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✿ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡s❝r❡✈❡r❡♠♦s ❛s á❧❣❡❜r❛s ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt ✭q✉❡ ❝❤❛♠❛r❡♠♦s s✐♠♣❧❡s♠❡♥t❡ ❞❡ á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt✮ ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r✲ ❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r E✳✳

      

    ✷✳✶ ❆çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r

    ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ 1

      , E , r, s) ❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E = (E ❝♦♥s✐st❡ ❞❡ ❝♦♥✲ 1 1

      , E → E ❥✉♥t♦s ♥ã♦ ✈❛③✐♦s E ❡ ❢✉♥çõ❡s r, s : E ✳ 1

      ❖s ❡❧❡♠❡♥t♦s ❞❡ E sã♦ ❝❤❛♠❛❞♦s ✈ért✐❝❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ E sã♦ ❞❡♥♦♠✐♥❛❞♦s ❛r❡st❛s✳ P❛r❛ ✉♠❛ ❛r❡st❛ e✱ r(e) é ♦ r❛♥❣❡ ❞❡ e ❡ s(e) é ♦ s♦✉r❝❡ ❞❡ e✳ 1 1

      , E , r, s} = {u, v, w, k}, E = ❊①❡♠♣❧♦ ✷✳✶✳✷✳ ❙❡❥❛ E = {E ✱ ❡♠ q✉❡ E {e, f, g, h}

      ✱ s(e) = w = r(h)✱ s(f) = u = r(e)✱ s(g) = v = r(f) ❡ s(h) = k = r(g) ✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❣r❛❢♦ ♣♦r f u // v e OO g oo w k h 1

      , E , r, s} = {u, v, w, k, z} ❊①❡♠♣❧♦ ✷✳✶✳✸✳ ❙❡❥❛ E = {E ✱ ❡♠ q✉❡ E ✱ 1 E = {f , f , f , f , f , f , f } ) ) = s(f ) = s(f ) = 1 2 3 4 5 6 7 ✱ z = s(f 7 ✱ u = r(f 1 1 2 s(f ) ) = s(f ) = s(f ) ) = s(f ) ) = 3 ✱ v = r(f 3 4 6 ✱ w = r(f 4 5 ❡ k = r(f 2 r(f ) = r(f ) = r(f ) 6 5 7 ✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❣r❛❢♦ ♣♦r f f f 3 4 1 // w // u v

      99 f 2 f 6 f 5  oo k z f 7

      ❉❡✜♥✐çã♦ ✷✳✶✳✹✳ ❙❡❥❛ E ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❡ s❡❥❛ K ✉♠ ❝♦r♣♦✳ ❆ K (E) á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❞❡ E ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ K✱ ❞❡♥♦t❛❞❛ ♣♦r L ✱

      } é ❛ K✲á❧❣❡❜r❛ ✉♥✐✈❡rs❛❧ ❣❡r❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ {v : v ∈ E ❞❡ 1

      : e ∈ E } ✐❞❡♠♣♦t❡♥t❡s ♦rt♦❣♦♥❛✐s ❡♥tr❡ s✐ ❡ ✉♠ ❝♦♥❥✉♥t♦ {e, e ❞❡ ❡❧❡♠❡♥t♦s s❛t✐s❢❛③❡♥❞♦✿ 1

      ✭✐✮ s(e)e = er(e) = e✱ ♣❛r❛ t♦❞♦ e ∈ E ❀ ∗ ∗ ∗ 1 = e s(e) = e

      ✭✐✐✮ r(e)e ✱ ♣❛r❛ t♦❞♦ e ∈ E ❀ 1 ∗ ∗ f = 0 e = r(e) ✭✐✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r e, f ∈ E t❡♠✲s❡ q✉❡ e s❡ e 6= f ❡ e ❀ X ee

      ✭✐✈✮ v = ✱ ♣❛r❛ t♦❞♦ ✈ért✐❝❡ v t❛❧ q✉❡ 0 < #{e : s(e) = e ∈E :s(e)=v 1 v} < ∞ ✳

      ❆❣♦r❛ s❡rã♦ ✐♥tr♦❞✉③✐❞❛s ❛s ❝♦♥✈❡♥çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ ❞❡✜♥✐r ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ q✉❡ é ✐s♦♠♦r❢♦ à á❧❣❡❜r❛ ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✲ ✈✐tt✳ ❉❡✜♥✐çã♦ ✷✳✶✳✺✳ ❯♠ ❝❛♠✐♥❤♦ ❞❡ t❛♠❛♥❤♦ n ❡♠ ✉♠ ❣r❛❢♦ E é ✉♠❛

      µ . . . µ n i ) = s(µ i ) s❡q✉ê♥❝✐❛ µ = µ 1 2 t❛❧ q✉❡ r(µ +1 ✱ ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n − 1}

      ✳ ❊s❝r❡✈❡♠♦s |µ| = n ♣❛r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡ µ✳ ❈❤❛♠❛✲ ♠♦s ♦s ✈ért✐❝❡s ❞❡ ❝❛♠✐♥❤♦s ❞❡ t❛♠❛♥❤♦ ③❡r♦✳ n ❉❡✜♥✐çã♦ ✷✳✶✳✻✳ E é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♠✐♥❤♦s ❞❡ t❛♠❛♥❤♦ n✳ n

      ❊st❡♥❞❡♠♦s ❛s ❛♣❧✐❝❛çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ ♣❛r❛ E ❞❡✜♥✐♥❞♦ s(µ) = s(µ ) ) 1 ❡ r(µ) = r(µ 1 ✱ s❡ n ≥ 2✱ ❡ s(v) = v = r(v)✱ ♣❛r❛ n = 0✳ ❉❡✜♥✐çã♦ ✷✳✶✳✼✳ W é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❡♠ E✱ ♦✉ s❡❥❛✱ [ 1 W = {µ . . . µ : µ ∈ E ) = s(µ ), ∀i ∈ {1, . . . , n − 1}}. 1 n i i i +1 n =1 ❡ r(µ

      ❉❡✜♥✐çã♦ ✷✳✶✳✽✳ W é ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ✐♥✜♥✐t♦s ❞❡ E✱ ♦✉ s❡❥❛✱ 1 W = {µ 1 µ 2 . . . : µ i ∈ E i ) = s(µ i +1 ), ∀i ∈ N}. 1 ∞ ❡ r(µ

      → E ∪ E ❆❣♦r❛ ✈❛♠♦s ❡st❡♥❞❡r s, r : E ♣❛r❛ W ∪ W ✳ ❉❡✜♥❛✿ s(µ) = s(µ 1 ) 1 µ 2 . . . ∈ W 1 µ 2 . . . µ n

      ✱ ♣❛r❛ µ = µ ♦✉ µ = µ ❀ r(ξ) = r(ξ n ) . . . ξ n ∈ W ✱ ♣❛r❛ ξ = ξ 1 ❀ r(v) = v = s(v)

      ✱ ♣❛r❛ v ∈ E ✳ . . . ξ n

      ❉❡✜♥✐çã♦ ✷✳✶✳✾✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝❛♠✐♥❤♦ ξ = ξ 1 . . . η m i = ξ i , 1 é ♦ ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ η = η s❡ m ≥ n ❡ η ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✵✳ ❖ ❣r❛✉ ❞❡ s❛í❞❛ ❞❡ ✉♠ ✈ért✐❝❡ v é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s e t❛✐s q✉❡ s(e) = v ✳ ❖ ❣r❛✉ ❞❡ ❡♥tr❛❞❛ ❞❡ ✉♠ ✈ért✐❝❡ v é ♦ ♥ú♠❡r♦ ❞❡ ❛r❡st❛s e t❛✐s q✉❡ r(e) = v

      ✳ ❉❡✜♥✐çã♦ ✷✳✶✳✶✶✳ ❯♠❛ ❢♦♥t❡ é ✉♠ ✈ért✐❝❡ ❝♦♠ ❣r❛✉ ❞❡ ❡♥tr❛❞❛ ③❡r♦✳ ❯♠ ♣♦ç♦ é ✉♠ ✈ért✐❝❡ ❝♦♠ ❣r❛✉ ❞❡ s❛í❞❛ ③❡r♦✳ 1

      ❙❡❥❛ F ♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r E ❡ s❡❥❛ X = {ξ ∈ W : r(ξ) é ✉♠ ♣♦ç♦}∪ {v ∈ E : v

      é ✉♠ ♣♦ç♦} ∪ W ✳ P❛r❛ ❝❛❞❛ c ∈ F✱ ❞❡✜♥❛✿

      X = X, ❡♠ q✉❡ 0 é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡ F❀

      X −1 = {ξ ∈ X : s(ξ) = r(b)} b ✱ ♣❛r❛ t♦❞♦ b ∈ W ❀ X a = {ξ ∈ X : ξ ξ . . . ξ = a} 1 2 |a| ✱ ♣❛r❛ t♦❞♦ a ∈ W ❀ −1

      X −1 = {ξ ∈ X : ξ ξ . . . ξ = a} = X a ∈ F ab 1 2 |a| ✱ ♣❛r❛ ab ✱ −1 ❝♦♠ a, b ∈ W , r(a) = r(b) ❡ ab ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ ♦✉ s❡❥❛✱ a 6= b |a| |b| ✱

      X c = ∅ ✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦ c ∈ F✳

      ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✷✳ P♦❞❡♠♦s ✈❡r W ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ F✳ b −1 Pr♦♣♦s✐çã♦ ✷✳✶✳✶✸✳ r(b) ∈ X s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r(b) é ✉♠ ♣♦ç♦✳ b = {r(b)} b = {b} −1 ❆❧é♠ ❞✐ss♦✱ s❡ r(b) é ✉♠ ♣♦ç♦ ❡♥tã♦ X ❡ X ✳ b −1 ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ r(b) ∈ X ✳ ❊♥tã♦ r(b) ∈ X ∩ E ❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ X✱ s❡❣✉❡ q✉❡ r(b) é ♣♦ç♦✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ r(b) é b −1 ♣♦ç♦✳ ❈♦♠♦ r(b) ∈ X ❡ r(b) = s(r(e))✱ t❡♠♦s q✉❡ r(b) ∈ X ✳ b −1

      ❙❡ r(b) é ♣♦ç♦✱ ❡♥tã♦✱ ♣❛r❛ ξ ∈ X ✱ r(b) = s(ξ) ✐♠♣❧✐❝❛ ❡♠ ξ ∈ E ∩ X b . . . ξ = b

      ✳ ❉♦♥❞❡ ξ = r(b)✳ P❛r❛ ξ ∈ X t❡♠♦s ξ 1 |b| ✱ ♦ q✉❡ ) = r(b) . . . ξ = b

      ✐♠♣❧✐❝❛ ❡♠ r(ξ |b| s❡r ♣♦ç♦✳ ▲♦❣♦✱ ξ = ξ 1 |b| ✳ v := {ξ ∈ X : ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✹✳ P❛r❛ t♦❞♦ v ∈ E ✱ t❡♠♦s v ∈ X s(ξ) = v} s❡✱ ❡ s♦♠❡♥t❡ s❡✱ v é ✉♠ ♣♦ç♦✳ ❉❡ ❢❛t♦✱ v ∈ X v ⇐⇒ v ∈ X ⇐⇒ v v = {v} é ♣♦ç♦ . ❆❧é♠ ❞✐ss♦✱ s❡ v é ✉♠ ♣♦ç♦✱ ❡♥tã♦ X ✳ ▲❡♠❛ ✷✳✶✳✶✺✳ ❙❡❥❛♠ a, c ∈ W , b, d ∈ W ∪ {0} ❡ v ∈ E ✳ ❊♥tã♦✿ (

      X −1 = X −1 a c ✱ s❡ r(a) = r(c) (1) X −1 ∩X −1 = a c ( ✱ ❝❛s♦ ❝♦♥trár✐♦

      X cd −1 −1 −1 ✱ s❡ r(a) = s(c) (2) X a ∩X cd =

      ✱ ❝❛s♦ ❝♦♥trár✐♦ (3)

      ❙✉♣♦♥❤❛ r(a) = r(b) ❡ r(c) = r(d)✳ −1 X ab −1 −1 ✱ s❡ a = ct, ♣❛r❛ ❛❧❣✉♠ t ∈ W ∪ {0}

      X ab ∩X cd = X −1 cd ✱ s❡ c = at, ♣❛r❛ ❛❧❣✉♠ t ∈ W ∪ {0} ( ✱ ❝❛s♦ ❝♦♥trár✐♦ X v = X −1 b ✱ s❡ r(b) = v

      (4) X v ∩X b −1 = ( −1 ✱ ❝❛s♦ ❝♦♥trár✐♦ X ab −1 ✱ s❡ s(a) = v

      (5) X v ∩X ab = [ ✱ ❝❛s♦ ❝♦♥trár✐♦ (6) X v = s (a)=v X −1 ab

      ❉❡♠♦♥str❛çã♦✿ (1) X −1 ∩ X −1 = {ξ ∈ X : s(ξ) = r(a)} ∩ {η ∈ X : s(η) = r(c)} a c

      = {ξ ∈ X : s(ξ) = r(a) = r(c)} ❊♥tã♦ (

      X −1 = X −1 a c ✱ s❡ r(a) = r(c) X −1 ∩ X −1 = a c −1 ✱ ❝❛s♦ ❝♦♥trár✐♦

      (2) ❙❡ r(c) = r(d) ❡ cd ❡stá ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ ❡♥tã♦✿

      X a −1 ∩ X cd −1 = {ξ ∈ X : s(ξ) = r(a)} ∩ {η ∈ X : η η . . . η = c} 1 2 |c| = {ξ ∈ X : s(ξ) = r(a) η . . . η = c}

      ❡ η 1 2 |c| = {ξ ∈ X : s(ξ) = r(a) = s(c) cd −1 = ∅ ❡ c é ❝♦♠❡ç♦ ♣❛r❛ ♦ ❝❛♠✐♥❤♦ ξ}

      ❙❡ r(c) 6= r(d)✱ ❡♥tã♦ X ✳ ▲♦❣♦✱ ( X −1 = X −1 cd c ✱ s❡ r(a) = s(c)

      X −1 ∩ X −1 = a cd −1 −1 ✱ ❝❛s♦ ❝♦♥trár✐♦ (3) X ab ∩ X cd = X a ∩ X c

      = {ξ ∈ X : ξ . . . x = a} ∩ {η ∈ X : η . . . η = c} 1 |a| 1 |c| = {ξ ∈ X : a

      é ❝♦♠❡ç♦ ❞❡ ξ ❡ c é ❝♦♠❡ç♦ ❞❡ ξ} ❚❡♠♦s três ❝❛s♦s✿

      ❝❛s♦ ✶✿ a ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ c ❡ c ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ a a ∩ X c = ∅. ✳ ❙❡♥❞♦ ❛ss✐♠✱ X

      ❝❛s♦ ✷✿ c é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ a✳ ◆❡st❡ ❝❛s♦✱ ❡①✐st❡ t ∈ W ∪ {0} t❛❧ q✉❡ a = ct✳ ❉❛í✱ X −1 ∩X −1 = X a ∩X c = {ξ ∈ X : a −1 . ab cd é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ ξ} = X ab ❝❛s♦ ✸✿ a é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ c✳ ◆❡st❡ ❝❛s♦✱ ❡①✐st❡ t ∈ W ∪ {0} t❛❧ q✉❡ c = at✳ ❉❛í✱ X −1 ∩X −1 = X ∩X = {ξ ∈ X : c −1 . b cd a c é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ ξ} = X cd −1

      (4) X v ∩ X b = {ξ ∈ X : s(ξ) = r(v) = v} ∩ {η ∈ X : s(η) = r(b)} = {ξ ∈ X : s(ξ) = v = r(b)}.

      ❊♥tã♦✱ ( X v = X b −1 −1 ✱ s❡ r(b) = v,

      X v ∩ X b = −1 = ∅ ✱ ❝❛s♦ ❝♦♥trár✐♦ ✭✺✮ ❙❡ r(a) 6= r(b)✱ ❡♥tã♦ X ab ✳ ❙❡ r(a) = r(b)✱ ❡♥tã♦✿ −1

      X v ∩ X ab = {ξ ∈ X : s(ξ) = v} ∩ {η ∈ X : a é ❝♦♠❡ç♦ ❞❡ η}

      = {ξ ∈ X : s(ξ) = v = s(a) ❡ a é ❝♦♠❡ç♦ ❞❡ ξ}. ▲♦❣♦✱ (

      X −1 ab ✱ s❡ s(a) = v X v ∩ X ab −1 =

      ✱ ❝❛s♦ ❝♦♥trár✐♦ (6) v η ⊆ [ ❙❡❥❛ ξ ∈ X ✱ ♦✉ s❡❥❛✱ s(ξ) = v✳ ❙❡ ξ ∈ W ∩ X✱ ❡♥tã♦ ξ ∈ X −1

      X ab : s (a)=v ✳ ❙❡ ξ ∈ {u ∈ E ✉ é ♣♦ç♦ }✱ ❡♥tã♦ s(ξ) = r(ξ) = v✳ ❙❡ [ ξ ∈ W ) = v ξ X −1

      ✱ ❡♥tã♦ s(ξ) = s(ξ 1 ❡✱ s❡♥❞♦ ❛ss✐♠✱ ξ ∈ X ab ✳ 1 s (a)=v ❆ ♦✉tr❛ ✐♥❝❧✉sã♦ s❡❣✉❡ ❞♦ ✐t❡♠ ✭✺✮✳ 1

      ❚❡♠♦s ❛té ❛❣♦r❛ ♦ ❣r✉♣♦ ❧✐✈r❡ F ❣❡r❛❞♦ ♣♦r E ❡ ♦ ❝♦♥❥✉♥t♦ X s♦❜r❡ ♦ q✉❛❧ F ✐rá ❛❣✐r✳ ❆✐♥❞❛ ❢❛❧t❛ ❞❡✜♥✐r♠♦s✱ ♣❛r❛ t♦❞♦ c ∈ F t❛❧ q✉❡ −1 −1 X c 6= ∅ c : X c → X c

      ✱ ❜✐❥❡çõ❡s θ ✳ : X → X

      ❉❡✜♥❛ θ ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡✳ P❛r❛ b ∈ W ✱ b : X b −1 → X b , b ❞❡✜♥❛ θ ♣♦r ξ 7→ bξ✳ ◆♦t❡ q✉❡ θ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ b b −1 ♣♦✐s✱ ♣❛r❛ ξ ∈ X t❡♠♦s q✉❡ s(ξ) = r(b) ❡✱ ♣♦rt❛♥t♦✱ bξ ∈ X ✳

      6= b −1 : X −1 → P❛r❛ a, b ∈ W ❝♦♠ r(a) = r(b) ❡ a |a| |b| ✱ ❞❡✜♥❛ θ ab ba

      X −1 , ξ . . . −1 ab ♣♦r ξ 7→ aξ |b|+1 |b|+2 ✳ ◆♦t❡ q✉❡ θ ab ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ −1 . . . ξ = b ♣♦✐s✱ ♣❛r❛ ξ ∈ X ba ✱ t❡♠♦s q✉❡ ξ 1 |b| ❡✱ ❝♦♠♦ r(a) = r(b) = r(ξ ) = s(ξ ) ξ . . . ∈ X −1 |b| |b|+1 ✱ s❡❣✉❡ q✉❡ aξ |b|+1 |b|+2 ab ✳ −1 : X → X −1 η . . .

      ❆✐♥❞❛✱ ❞❡✜♥❛ θ b b b ✱ ♣♦r η 7→ η ✱ s❡ r(b) |b|+1 |b|+2 ♥ã♦ é ♣♦ç♦✱ ❡ η 7→ r(b)✱ s❡ r(b) é ♣♦ç♦✳ ba : X ab → X ba η . . . −1 −1 −1

      P♦r ✜♠✱ ❞❡✜♥❛ θ ✱ ♣♦r η 7→ bη |a|+1 |a|+2 ✳ b = θ −1 Pr♦♣♦s✐çã♦ ✷✳✶✳✶✻✳ P❛r❛ b ∈ W ✱ t❡♠♦s θ −1 ✳ P❛r❛ a, b ∈ −1 b W 6= b = θ −1

      ❝♦♠ r(a) = r(b) ❡ a |a| |b| ✱ t❡♠♦s θ −1 ba ✳ b −1 ab ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ η ∈ X ✱ s❡❣✉❡ q✉❡ −1

      (θ −1 ◦ θ b )(η) = θ (η η . . .) = bη η . . . = η, b |b|+1 |b|+2 |b|+1 |b|+2 b s❡ r(b) ♥ã♦ é ♣♦ç♦✳ ❙❡ r(b) é ♣♦ç♦✱ s❡❣✉❡ q✉❡ (θ b −1 ◦ θ b )(η) = θ −1 b (r(b)) = br(b) = η. ❉❛❞♦ ξ ∈ X b ✱ s❡❣✉❡ q✉❡ (θ b ◦ θ b −1 )(ξ) = θ b (bξ) = ξ

      P❛r❛ ξ ∈ X ba −1 ✱ s❡❣✉❡ q✉❡ (θ ba −1 ◦ θ ab −1 )(ξ) = (θ ba −1 )(aξ |b|+1 ξ |b|+2 . . .) = bξ |b|+1 ξ |b|+2 . . . = ξ. P❛r❛ η ∈ X ab −1 s❡❣✉❡ q✉❡

      ❛✐♥❞❛ q✉❡ s❡ g = 0 ♦✉ h = 0 ❡♥tã♦ ❛ ✐♥❝❧✉sã♦ é t❛♠❜é♠ tr✐✈✐❛❧✳ ❊♥tã♦ s✉♣♦♥❤❛ q✉❡ g = ab −1 ❡ h = cd −1 ❝♦♠ a, b, c, d ∈ W ∪

      ♣♦✐s ♥❡♥❤✉♠ ξ ∈ X ♣♦❞❡ ❝♦♠❡ç❛r ❝♦♠ a ❡ d s✐♠✉❧t❛♥❡❛♠❡♥t❡✳ ❝❛s♦ ✷✿ a é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ d✱ ❡ ❡s❝r❡✈❡♠♦s d = ad ✳ ◆♦t❡ q✉❡ a 6= 0 ⇒ d 6= 0✳ ❉❛í✱

      ❡ ❝♦♠♦ a ♥ã♦ é ❝♦♠❡ç♦ ❞❡ d ❡♥tã♦ d /∈ X a ✳ ❉❛í✱ X g ∩ X h −1 = X ab −1 ∩ X (cd −1 ) −1 = X a ∩ X d = ∅,

      ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ a 6= 0 ❡ d 6= 0✳ ❈♦♠♦ d ♥ã♦ é ❝♦♠❡ç♦ ❞❡ a✱ ❡♥tã♦ a / ∈ X d

      ✳ ❚❡♠♦s ✸ ❝❛s♦s✳ ❝❛s♦ ✶✿ a ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ d ❡ d ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ a

      ✭✐✐✮ ❱❛♠♦s ♠♦str❛r q✉❡ θ −1 h (X h ∩ X g −1 ) ⊆ X (gh) −1

      , cd −1 ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ r(a) = r(b)✱ r(c) = r(d)✱ a, b ♥ã♦ s✐♠✉❧t❛♥❡❛♠❡♥t❡ 0 ❡ ♥❡♠ c, d s✐♠✉❧t❛♥❡❛♠❡♥t❡ 0✳

      {0} ✱ ab −1

      : a, b ∈ W ∪ {0}} ∪ {0} é ❢❡❝❤❛❞♦ ♣♦r ✐♥✈❡rs♦s✳ ◆♦t❡

      (θ ab −1 ◦ θ ba −1 )(η) = (θ ab −1 )(bη |a|+1 η |a|+2 . . .) = aη |a|+1 η |a|+2 . . . = η. Pr♦♣♦s✐çã♦ ✷✳✶✳✶✼✳ ❖ ♣❛r θ = ({X c } c

      ✐♥❝❧✉sã♦ ♦✉ ✐❣✉❛❧❞❛❞❡ é tr✐✈✐❛❧✳ ◆♦t❡ q✉❡ {ab −1

      {ab −1 : a, b ∈ W ∪ {0}} ∪ {0} ✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱ X g ∩ X h −1 = ∅ ❡ ❛

      ❉❡♠♦♥str❛çã♦✿ ❖ ❛①✐♦♠❛ ✭✐✮ ❞❡ s❡❣✉❡ ❞❛s ❞❡✜♥✐çõ❡s ❞❡ X ❡ θ ✳ P❛r❛ ♦s ♦✉tr♦s ❛①✐♦♠❛s ❞❡ ❜❛st❛ q✉❡ ✈❡r✐✜q✉❡♠♦s ♣❛r❛ g, h ∈

      ) é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳

      ∈F

      , {θ c } c

      ∈F

      X g ∩ X h −1 = X ab −1 ∩ X (cd −1 ) −1

      =

      X g −1 h −1 = X bc −1 = X b . ▲♦❣♦✱

      ✱ ❡♥tã♦

      ✳ ❈♦♠♦ X g −1 h −1 = X ba −1 dc −1 = X c −1

      ✳ ▲♦❣♦✱ X c −1 = X a −1

      = 0 ✱ ❡♥tã♦ a = d ❡ r(c) = r(d) = r(a)

      ❝♦♠♦ q✉❡rí❛♠♦s✳ ❊♠ (∗∗∗)✱ t❡♠♦s ♠❛✐s ❞♦✐s ❝❛s♦s✳ ❙❡ b = d

      ▲♦❣♦✱ θ b (ξ) ∈ X bd ✳ ❆ss✐♠✱ θ g −1 (X g ∩ X h −1 ) = θ b (X b −1 ∩ X d ) ⊆ X bd = X g −1 h −1 ,

      ∩ X d t❡♠♦s ξ = dξ ❡ t❛♠❜é♠ θ g −1 (ξ) = θ ba −1 (ξ) = θ b (ξ) = bdξ = (bd)ξ .

      X g −1 h −1 = X bdc −1 = X bd . ❉❛í✱ ♣❛r❛ t♦❞♦ ξ ∈ X b −1

      ▲♦❣♦✱ bd 6= 0 ❡ ❡♥tã♦

      ❊♠ (∗∗)✱ t❡♠♦s q✉❡ g −1 h −1 = bdc −1 ❡ d 6= 0, b 6= 0✱ ♣♦✐s a = 0✳

      θ −1 g (X g ∩ X h −1 ) = θ b (X b −1 ∩ X c −1 ) ⊆ X b = X g −1 h −1 , ❝♦♠♦ q✉❡rí❛♠♦s✳

      ❡ b 6= 0✱ ♣♦✐s a = 0✳ ❊♥tã♦✱

      X b −1 ∩ X c −1 ✱ s❡ a = d = 0

      ✱ s❡ a 6= 0 ✭✯✯✯✮ ❊♠ (∗)✱ t❡♠♦s q✉❡ g −1 h −1 = ba −1 dc −1 = bc −1

      ✱ s❡ a = d ❡ d 6= 0 ✭✯✯✮ θ ba −1 (X d )

      ✱ s❡ a = d = 0 ✭✯✮ θ b (X b −1 ∩ X d )

      = θ b (X b −1 ∩ X c −1 )

      ❊♥tã♦✱ θ −1 g (X g ∩ X −1 h ) = θ ba −1 (X g ∩ X h −1 )

      X d ✱ s❡ a 6= 0

      X b −1 ∩ X d ✱ s❡ a = 0 ❡ d 6= 0

      X b −1 ∩ X c −1 ✱ s❡ a = d = 0

      =

      X a ∩ X d ✱ s❡ a 6= 0

      X b −1 ∩ X d ✱ s❡ a = 0 ❡ d 6= 0

      θ g −1 (X g ∩ X h −1 ) = θ ba −1 (X d ) = θ a −1 (X a ) = X a −1 = X c −1 = X g −1 h −1 .

      ❙❡ bd 6= 0

      θ ba −1 (η) ✱ ♣❛r❛ ❛❧❣✉♠ η ∈ X ab −1 ∩ X dc −1 ✳ ❆ss✐♠✱ η = dη

      (X c ) ♦ s✉❜❝♦♥❥✉♥t♦ ❞❡ F(X) q✉❡ ❝♦♥té♠ ❛♣❡♥❛s ❛ ❢✉♥çã♦ ♥✉❧❛✳

      = ∅ ✱ s❡❥❛ F

      K ✲á❧❣❡❜r❛ ❞❡ ❢✉♥çõ❡s ❞❡ X c ❡♠ K✳ P❛r❛ c ∈ F✱ ❝♦♠ X c

      θ hg (ξ) = θ c (bd ) −1 ((bd ) = cη . ❝❛s♦ ✸✿ d é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ a✱ ❡ ❡s❝r❡✈❡♠♦s a = da . ❊st❡ ❝❛s♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ ❚❡♠♦s ❡♥tã♦ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ♥♦ ♥í✈❡❧ ❞❡ ❝♦♥❥✉♥t♦s✳ ❙❡❥❛ F(X) ❛ K✲á❧❣❡❜r❛ ❞❛s ❢✉♥çõ❡s ❞❡ X ♥♦ ❝♦r♣♦ K✱ ❝♦♠ ❛ ♠✉❧✲ t✐♣❧✐çã♦ ♣♦♥t♦ ❛ ♣♦♥t♦✳ P❛r❛ ❝❛❞❛ c ∈ F✱ ❝♦♠ X c 6= ∅ ✱ s❡❥❛ F(X c )

      θ h (θ g (ξ)) = θ cd −1 (θ ab −1 (bd η )) = θ cd −1 (ad η ) = θ cd −1 (dη ) = cη P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❣✉❡ q✉❡

      ▲♦❣♦✱ ξ = θ ba −1 (η) = θ ba −1 (ad ξ ) = bd η . ❉❛í✱ ♣♦r ✉♠ ❧❛❞♦✱ t❡♠♦s q✉❡

      = ad η

      (X g ∩ X h −1 ) = θ ba −1 (X ab −1 ∩ X dc −1 ) ✳ ❊♥tã♦ ξ =

      ✱ ❡♥tã♦

      ❝❛s♦ ✷✿ a é ❝♦♠❡ç♦ ❞❛ ♣❛❧❛✈r❛ d✱ ❡ ❡s❝r❡✈❡♠♦s d = ad ✳ ❙❡❥❛ ξ ∈ θ −1 g

      = ∅ ✱ ❝♦♠♦ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❞♦ ❛①✐♦♠❛ ✭✐✐✮✳

      ✳ ◆❡st❡ ❝❛s♦✱ X g ∩ X h −1

      ❝❛s♦s✳ ❝❛s♦ ✶✿ a ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ d ❡ d ♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ a

      (X h ∩ X g −1 ) ✳ ❚❡♠♦s ✸

      θ ba −1 (ξ) = θ ba −1 (ad ξ ) = bd ξ ∈ X bd . ❝❛s♦ ✸✿ d é ❝♦♠❡ç♦ ❞❛ ♣❛❧❛✈r❛ a✱ ❡ ❡s❝r❡✈❡♠♦s a = da ✳ ❊st❡ ❝❛s♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ ✭✐✐✐✮ θ g ◦ θ h (ξ) = θ gh (ξ) ✱ ♣❛r❛ t♦❞♦ ξ ∈ θ h −1

      = ad ξ ❡ ❡♥tã♦

      X g −1 h −1 = X ba −1 dc −1 = X ba −1 ad c −1 = X bd c −1 = X bd . ❚❡♠♦s q✉❡ ♠♦str❛r q✉❡ θ g −1 (X d ) ⊆ X bd ✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ ξ ∈ X d ✱ ❡s❝r❡✈❡♠♦s ξ = dξ

      P❛r❛ t♦❞♦ c ∈ F✱ F(X c ) é ✉♠ ✐❞❡❛❧ ♥❛ K✲á❧❣❡❜r❛ F(X)✳ ❆❣♦r❛✱ ♣❛r❛ ❝❛❞❛ c ∈ F✱ ❞❡✜♥❛

      F F −1 α c : (X c ) → (X c ) −1 f 7→ f ◦ θ c c

      P❛r❛ t♦❞♦ c ∈ F✱ α é ✉♠ K✲✐s♦♠♦r✜s♠♦✳ ❊✱ ♣❡❧♦ ❊①❡♠♣❧♦ c )} c , {α c } c ) s❡❣✉❡ q✉❡ α = ({F(X é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ F s♦❜r❡ F ∈F ∈F (X)

      ✳ ❖ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡st❛ ❛çã♦ é ✏♠✉✐t♦ ❣r❛♥❞❡✑

      ♣❛r❛ ♦s ♥♦ss♦s ♣r♦♣ós✐t♦s✳ ◗✉❡r❡♠♦s ✏❞✐♠✐♥✉✐r✑ ♥♦ss❛ á❧❣❡❜r❛✳ c ∈ F(X c ) c P❛r❛ ✐ss♦✱ s❡❥❛ 1 ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠ X ✱ ♦✉ s❡❥❛✱ ( 1 c

      ✱ s❡ ξ ∈ X 1 c (ξ) = 1 X c (ξ) = [ξ ∈ X c ] = c v ∈ F(X v ) v ✱ s❡ ξ /∈ X P❛r❛ ❝❛❞❛ v ∈ E ✱ s❡❥❛ 1 ❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠ X ✳

      ▲❡♠❛ ✷✳✶✳✶✽✳ ❙❡❥❛♠ p, q ∈ F✳ ❊♥tã♦✿ p (1 p −1 1 q ) = 1 p 1 pq

      ✭✐✮ α ❀ ✭✐✐✮ P❛r❛ a ∈ W, b ∈ W ∪ {0}✱ s❡❣✉❡ q✉❡ ( 1 a

      ✱ s❡ r(a) = v α a (1 −1 a 1 v ) =

      ✱ s❡ r(a) 6= v ❡ ( 1 −1 ab ✱ s❡ s(b) = v

      α −1 (1 −1 ab ba 1 v ) = −1 −1 ✱ s❡ s(b) 6= v ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛ p = ab ❡ q = cd ✱ ❝♦♠ a, c ∈ W ❡

      b, d ∈ W ∪ {0} ✱ r(a) = r(b) ❡ r(c) = r(d)✳

      Pr✐♠❡✐r♦ s✉♣♦♥❤❛ q✉❡ c = at✱ ❝♦♠ t ∈ W ∪ {0}✳ ❙❡❥❛ ξ ∈ X✳ ❊♥tã♦✿

      α p (1 p −1 1 q )(ξ) = (α p (1 p −1 )α p (1 q ))(ξ) = α p (1 p −1 )(ξ)α p (1 q )(ξ) = 1 p −1 (θ p −1 (ξ))1 q (θ p −1 (ξ)) −1 −1 −1 −1 = 1 ab (θ ab (ξ))1 atd (θ ab (ξ)) −1 −1 −1 −1 = [θ ab ∈ X ab ][θ ab (ξ) ∈ X atd ] −1 −1 −1 = [ξ ∈ X ba ][θ ab (ξ) ∈ X atd ] −1 = [ξ ∈ X btd ]

      1 p 1 pq (ξ) = 1 p (ξ)1 pq (ξ) = [ξ ∈ X p ][ξ ∈ X pq ] = [ξ ∈ X ba −1 ∩ X ba −1 cd −1 ] = [ξ ∈ X ba −1 ∩ X btd −1 ] = [ξ ∈ X btd −1 ]

      ❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ a = ct✱ ♣❛r❛ ❛❧❣✉♠ t ∈ W ✱ ♠♦str❛✲s❡ q✉❡ ♦ r❡s✉❧t❛❞♦ s❡❣✉❡✳ ❙❡ a 6= ct✱ ♣❛r❛ t♦❞♦ t ∈ W ✱ ❡ c 6= at✱ ♣❛r❛ t♦❞♦ t ∈ W ✱ ❡♥tã♦✿

      1 ab −1

      1 cd −1 = 0 ✱ ♣❡❧♦ ❧❡♠❛ ✸✳✶✳✶

      ❡

      1 ba −1 cd −1 = 0 ✱ ♣❡❧❛ ❞❡✜♥✐çã♦✳

      ❆s ♦✉tr❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ p, q ∈ F sã♦ s✐♠✐❧❛r❡s✳ ✭✐✐✮ ❙❡❥❛♠ a ∈ W, b ∈ W ∪ {0}✳ P❛r❛ t♦❞♦ ξ ∈ X✱ t❡♠♦s✿

      α a (1 a −1 1 v )(ξ) = α a (1 a −1 )(ξ)α a (1 v )(ξ) = 1 a −1 (θ a −1 )(ξ)1 v (θ a −1 )(ξ) = [θ a −1 (ξ) ∈ X a −1 ][θ a −1 (ξ) ∈ X v ] = [ξ ∈ X a ][ξ ∈ X v ∩ X a ] = [ξ ∈ X v ∩ X a ] = ( 1 a

      ✱ s❡ X v = X a ✱ s❡ r(a) 6= v

      α ab −1 (1 ab −1 1 v )(ξ) = α ab −1 (1 ab −1 )(ξ)α ab −1 (1 v )(ξ) = 1 ba −1 (θ ba −1 )(ξ)1 v (θ ba −1 )(ξ) = [θ ba −1 (ξ) ∈ X ba −1 ][θ ba −1 (ξ) ∈ X v ] = [ξ ∈ X ab −1 ][ξ ∈ X ab −1 ∩ X v ] = ( 1 ab −1

      ✱ s❡ X v ∩ X ab −1 = X ab −1 ✭ s❡ s(a) = v) ✱ s❡ s(a) 6= v

      ❆❣♦r❛ ♣♦❞❡♠♦s ❞❡✜♥✐r ❛ ❛çã♦ ♣❛r❝✐❛❧ q✉❡ ✐♥❞✉③ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ q✉❡ é ✐s♦♠♦r❢♦ à á❧❣❡❜r❛ ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt✳

      = span {1 p : p ∈ F \ {0}} ∪ {1 v : v ∈ E } ❙❡❥❛ D(X) = D ✭s♣❛♥

      ❑✲❧✐♥❡❛r✮✳ p ⊆ F(X p ) ❙❡❥❛✱ ♣❛r❛ ❝❛❞❛ p ∈ F \ {0}✱ D ❞❡✜♥✐❞♦ ❝♦♠♦

      D p = 1 p D = span{1 p p p 1 q : q ∈ F}. ❖❜s❡r✈❛çã♦ ✷✳✶✳✶✾✳ ◆♦t❡ q✉❡ D(X) ❡ D sã♦ ❑✲á❧❣❡❜r❛s ❡ q✉❡ D é ✉♠ ✐❞❡❛❧ ❞❡ D ✱ ♣❛r❛ t♦❞♦ p ∈ F✳ p (1 p −1 1 q ) = 1 p

      1 pq ❈♦♠♦ α ✱ ❝♦♥s✐❞❡r❡✱ ♣❛r❛ ❝❛❞❛ p ∈ F✱ ❛ r❡str✐çã♦ p p −1

      ❞❡ α ❛ D ✿ α p : D −1 → D p p p } p , {D p } p ) 1 p −1 1 q 7→ α p (1 p −1 1 q ) = 1 p 1 pq

      ❊♥tã♦ α = ({α é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳ α F ∈F ∈F ❙❡❥❛ D(X) ⋊ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ α✳

      ✷✳✷ ➪❧❣❡❜r❛s ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt

      ◆❡st❛ s❡çã♦✱ ❞❡s❝r❡✈❡r❡♠♦s ❛ á❧❣❡❜r❛ ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt ❛ss♦✲ ❝✐❛❞❛ ❛ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E ❝♦♠♦ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧✳ K (E) α F Pr♦♣♦s✐çã♦ ✷✳✷✳✶✳ ❊①✐st❡ ✉♠ K✲❤♦♠♦♠♦r✜s♠♦ ϕ ❞❡ L ❡♠ D(X)⋊ 1 ∗ e δ e ) = 1 e δ e −1 −1 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ e ∈ E ✱ ϕ(e) = 1 ❡ ϕ(e ❡✱ ♣❛r❛ v δ t♦❞♦ v ∈ E ✱ ϕ(v) = 1 ✳ e δ e , 1 e −1 δ e −1 : e ∈ E } v δ : 1

      ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s {1 ❡ {1 F v ∈ E } α ❡♠ D(X) ⋊ ✳

      ❱❛♠♦s ♠♦str❛r q✉❡ t❛✐s ❝♦♥❥✉♥t♦s s❛t✐s❢❛③❡♠ ❛s r❡❧❛çõ❡s q✉❡ ❞❡✜✲ ♥❡♠ ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt✳ s δ 1 e δ e = α (α (1 s )1 e )δ e = 1 s

      1 e δ e = 1 e δ e ✭✐✮ 1 (e) (e) (e) ✱ ♣♦✐s 1 s (e) (e) (e) 1 e δ e (ξ) = 1 X s ∩X e (ξ) = [ξ ∈ X s ∩X e ] = [ξ ∈ X e ] = 1 e (ξ). −1 −1 1 e δ e

      1 r δ = α e (α e (1 e )1 r )δ e = α e (1 e (e) (e) (e) 1 r )δ e = 1 e 1 er δ e = 1 e (e) 1 e δ e = 1 e δ e

      ✭✐✐✮ 1 r δ (e) (e) (e) 1 e −1 δ e −1 = α (α (1 r )1 e −1 )δ e −1 = 1 r 1 e −1 δ e −1 = 1 e −1 δ e −1 , ♣♦✐s −1 1 r (e) (e) e−1 (e) e−1 1 e (ξ) = 1 X r

      1 X (ξ) = 1 X r ∩X = [ξ ∈ X r ∩ X e −1 ] = [ξ ∈ X e −1 ] (e) −1 −1 −1 −1 −1 −1 −1 = 1 e −1 1 e δ e

      1 s δ = α e (α e (1 e )1 s )δ e = α e (1 e (e) (e) (e) −1 −1 −1 −1 −1 −1 1 s )δ e = 1 e −1 −1 1 e s δ e = 1 e (e) 1 e δ e 1 = 1 e δ e .

      ✭✐✐✐✮ ❙❡❥❛♠ f, g ∈ E ✳ ❊♥tã♦ −1 −1 −1 −1 −1 −1 −1 1 f δ f 1 g δ g = α f (α f (1 f )1 g )δ f g = α f (1 f f −1 1 g )δ f g .

      1 g = 0 ❙❡ f 6= g✱ ❡♥tã♦ ♣❡❧♦ ❧❡♠❛ ✸✳✶✳✶✱ 1 ✳ ▲♦❣♦✱ −1 −1 1 f δ f

      1 g δ g = 0 ✱ s❡ f 6= g. ❙❡ f = g✱ ❡♥tã♦ 1 −1 δ −1 f f f f f f f 1 f δ f = α −1 (α f (1 −1 )1 f )δ −1 = α −1 (1 f 1 f )δ f = 1 r −1 = α −1 (1 f )δ = 1 −1 δ f f

      ❈♦♠♦ 1 (e) ✱ ♣❡❧♦ ❧❡♠❛ ✸✳✶✳✶✱ s❡❣✉❡ q✉❡ −1 −1 1 f δ f 1 f δ f = 1 r δ . (e)

      ✭✐✈✮ ❙❡❥❛ v ∈ E t❛❧ q✉❡ 0 < #{e : s(e) = v} < ∞✳ ▲❡♠❜r❡ q✉❡ [ X v = X e X e ∈E :s(e)=v 1 −1 −1 ✳ ❊♥tã♦✿ X X e ∈E :s(e)=v e ∈E :s(e)=v e ∈E :s(e)=v 1 e δ e , 1 −1 δ −1 : e ∈ E } v δ : v ∈ E } 1 e δ e 1 e δ e = 1 1 1 e δ = ( 1 1 e )δ = 1 v δ .

      ❈♦♠♦ {1 e e ❡ {1 s❛t✐s❢❛③❡♠ ❛s ❝♦♥❞✐çõ❡s ❞❛ ➪❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt✱ ♣❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧✱ ❡①✐st❡ ✉♠ K (E) → D(X) ⋊ α F 1

      ❤♦♠♦♠♦r✜s♠♦ ϕ : L t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ e ∈ E ✱ ϕ(e) = 1 e δ e ) = 1 −1 δ −1 v δ

      ❡ ϕ(e e e ❡✱ ♣❛r❛ t♦❞♦ v ∈ E ✱ ϕ(v) = 1 ✳

      ❆♥t❡s ❞❡ ♠♦str❛r♠♦s ❛ ✐♥❥❡t✐✈✐❞❛❞❡ ❞❡ ϕ✱ s❡❣✉✐♠♦s ❝♦♠ ❛❧❣✉♠❛s ❞❡✜♥✐çõ❡s✳ ❉❡✜♥✐çã♦ ✷✳✷✳✷✳ ❙❡❥❛ A ✉♠ ❛♥❡❧✳ ❉✐③❡♠♦s q✉❡ A é Z✲❣r❛❞✉❛❞♦ s❡ n } n ❡①✐st❡ ✉♠❛ ❝♦❧❡çã♦ ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s {A ❞❡ A t❛❧ q✉❡ M ∈Z

      A n ✭✐✮ A = ❀ n m A n ⊆ A m ∈Z

      ✭✐✐✮ A +n ✱ ♣❛r❛ q✉❛✐sq✉❡r m, n ∈ Z✳ ❉❡✜♥✐çã♦ ✷✳✷✳✸✳ P❛r❛ ❝❛❞❛ p ∈ F✱ s❡❥❛ |p| := m − n✱ ❡♠ q✉❡ m é ♦ 1

      ♥ú♠❡r♦ ❞❡ ❣❡r❛❞♦r❡s ✭❡❧❡♠❡♥t♦s ❞❡ E ✮ ❞❡ p ❡ n é ♦ ♥ú♠❡r♦ ❞❡ ✐♥✈❡rs♦s ❞❡ ❣❡r❛❞♦r❡s ❞❡ p✳

      P❛r❛ ♠♦str❛r q✉❡ ♦ K✲❤♦♠♦♠♦r✜s♠♦ ❞❡ é ✐♥❥❡t✐✈♦ ✉s❛r❡♠♦s ♦ ❚❡♦r❡♠❛ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❣r❛❞✉❛çã♦ ❞❡ ❬✼❪ q✉❡ ❛✜r♠❛ q✉❡ 1

      , E , r, s) K (E) ❚❡♦r❡♠❛ ✷✳✷✳✹✳ ❙❡❥❛ E = (E ✉♠ ❣r❛❢♦ ❡ s❡❥❛ L ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❛ss♦❝✐❛❞❛ ❛ E ❝♦♠ ❛ ❣r❛❞✉❛çã♦ ✉s✉❛❧✳ ❙❡ A é ✉♠ K (E) → A ❛♥❡❧ Z✲❣r❛❞✉❛❞♦ ❡ ϕ : L é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ❞❡ ❛♥❡❧ t❛❧ q✉❡ ϕ(v) 6= 0✱ ♣❛r❛ t♦❞♦ v ∈ E ✱ ❡♥tã♦ ϕ é ✐♥❥❡t✐✈❛✳ α F

      P❛r❛ ✐ss♦✱ ✉s❛r❡♠♦s ❛ s❡❣✉✐♥t❡ Z✲❣r❛❞✉❛çã♦ ♣❛r❛ D(X) ⋊ ✭✈❡❥❛ z ⊆ D(X) ⋊ α F ❬✼❪✮✿ ♣❛r❛ ❝❛❞❛ z ∈ Z✱ ❞❡✜♥❛ A ❝♦♠♦ ♦ s♣❛♥ ❑✲❧✐♥❡❛r p δ p : a p ∈ D p K (E) Z ❞❡ {a ❡ |p| = z}✳ P❛r❛ L ✱ ❡st❛♠♦s ❝♦♥s✐❞❡r❛♥❞♦ ❛ z } z

      ✲❣r❛❞✉❛çã♦ {R ∈Z ✱ ❡♠ q✉❡ −1 R z = span{ab : a, b ∈ W ∪ E , |a| − |b| = z}. K (E) → D(X) ⋊ α F ❚❡♦r❡♠❛ ✷✳✷✳✺✳ ❖ ❤♦♠♦♠♦r✜s♠♦ ϕ : L ❞❛ ♣r♦✲ ♣♦s✐çã♦ é ✉♠ ❑✲✐s♦♠♦r✜s♠♦✳ ❉❡♠♦♥str❛çã♦✿ ❱❛♠♦s ♠♦str❛r q✉❡ ϕ é ✐♥❥❡t♦r❛✳ P❡❧♦ t❡♦r❡♠❛ ❞❛ ✉♥✐❝✐❞❛❞❡ ❞❡ ❣r❛❞✉❛çã♦✱ só é ♣r❡❝✐s♦ ♠♦str❛r q✉❡ ϕ é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❛♥❡❧ ❣r❛❞✉❛❞♦ t❛❧ q✉❡ ϕ(v) 6= 0✱ ♣❛r❛ t♦❞♦ v ∈ E ✳ −1 −1

      ∈ R z ) ∈ P❛r❛ ❝❛❞❛ ab ✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ϕ✱ t❡♠♦s q✉❡ ϕ(ab −1 −1

      D ab δ ab ✳ −1 −1 −1

      | = |a|−|b| = z ab δ ab ⊆ A z z ) ⊆ ❈♦♠♦ |ab ✱ ❡♥tã♦ D ✳ ▲♦❣♦✱ ϕ(R

      A z ✳ v δ 6= 0 v 6= 0

      ❆❧é♠ ❞✐ss♦✱ ϕ(v) = 1 ✱ ♣♦✐s 1 ♣❛r❛ t♦❞♦ v ∈ E ✳ ❊♥tã♦✱ s❡❣✉❡ q✉❡ ϕ é ✐♥❥❡t♦r❛✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ ϕ é s♦❜r❡❥❡t♦r❛✳ a δ a ) = ❆✜r♠❛çã♦✿ P❛r❛ t♦❞♦ a ∈ W ✱ t❡♠♦s q✉❡ ϕ(a) = 1 ❡ ϕ(a 1 −1 δ −1 a a ✳ ❆❧é♠ ❞✐ss♦✱ ♣❛r❛ q✉❛✐sq✉❡r a, b ∈ W ❝♦♠ a |a| |b| ❡ r(a) = 6= b r(b) ) = 1 −1 δ −1

      ✱ s❡❣✉❡ q✉❡ ϕ(ab ab ab ✳

      p δ p ∈

      ◆♦t❡ q✉❡✱ ❞❛ ❛✜r♠❛çã♦✱ s❡❣✉❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ p ∈ F \ {0}✱ 1 Im(ϕ)

      ✳ 1 . . . a n ∈ W ❙❡❥❛ a = a ✳ ❙❡ n = 1✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡ ϕ✱ s❡❣✉❡ a δ a 2 . . . a n ) = q✉❡ ϕ(a) = 1 ✳ ❙✉♣♦♥❤❛ n ≥ 2 ❡ q✉❡ s❡❥❛ ✈á❧✐❞♦ ϕ(a

      1 δ a ...a n a ...a n ✳ ❊♥tã♦ 2 2 ϕ(a) = ϕ(a )ϕ(a . . . a n ) 1 2

      = 1 a δ a 1 1 1 a ...a δ a ...a 2 n 2 n = α a (α −1 (1 a )1 a ...a )δ a a ...a 1 a 1 2 n 1 2 n

      = α a (1 −1 1 a 1 1 a ...a )δ a 2 n = 1 a 1 1 a ...a n δ a = 1 a ...a n δ a 1 1 = 1 a δ a .

      ) = 1 a −1 δ a −1 ❆♥❛❧♦❣❛♠❡♥t❡✱ ϕ(a ✳

      6= a ❙❡❥❛♠ a, b ∈ W ❝♦♠ r(a) = r(b) ❡ a |a| |b| ✳ ❊♥tã♦✿ −1

      ϕ(ab ) = 1 a δ a 1 −1 δ −1 = α a (α −1 (1 a )1 −1 )δ −1 b b a b ab = α a (1 −1 a b ab ab ab 1 −1 )δ −1 = 1 a 1 −1 δ −1 = 1 −1 δ −1 . ab ab p δ p ⊆

      P❛r❛ ♠♦str❛r q✉❡ ϕ é s♦❜r❡❥❡t♦r❛✱ é s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ D Im(ϕ)

      ✱ ♣❛r❛ t♦❞♦ p ∈ F✳ δ ⊆ Im(ϕ)

      Pr✐♠❡✐r♦✱ ✈❛♠♦s ♠♦str❛r q✉❡ D ✳ P♦r ❧✐♥❡❛r✐❞❛❞❡✱ é v δ ∈ Im(ϕ) s✉✜❝✐❡♥t❡ ♠♦str❛r q✉❡ 1 ❡ q✉❡✱ ♣❛r❛ ❝❛❞❛ p ∈ F \ {0}✱ 1 p δ ∈ Im(ϕ)

      ✳ v δ = ϕ(v) ∈ Im(ϕ) ❙❛❜❡♠♦s q✉❡ 1 ✳ ❆❣♦r❛✱ s❡❥❛ p ∈ F \ {0}✳ ◆♦t❡ q✉❡✿

      1 p δ p 1 p −1 δ p −1 = α p (α p −1 (1 p )1 p −1 )δ = α p (1 p −1 1 p −1 )δ = α p (1 p −1 )δ p δ = 1 p δ p −1 −1 −1 −1 = 1 p δ .

      1 p δ p p δ p , 1 p δ p ∈ Im(ϕ) ❖✉ s❡❥❛✱ 1 ✳ ❈♦♠♦ 1 ✱ ❡♥tã♦ 1 p δ ∈ Im(ϕ)

      ✳ p δ p ∈ Im(ϕ) ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ D ✳ P♦r ❧✐♥❡❛r✐❞❛❞❡✱ é s✉✜❝✐✲ p 1 q δ q ∈ Im(ϕ)

      ❡♥t❡ ♠♦str❛r q✉❡ 1 ✱ ♣❛r❛ q✉❛✐sq✉❡r p, q ∈ F \ {0}✳ ◆♦t❡ q✉❡✿ 1 p δ 1 q δ q = α (α (1 p )1 q )δ q = 1 p 1 q δ q .

      p δ , 1 q δ q ∈ Im(ϕ) p

      1 q δ q ∈ Im(ϕ) ❈♦♠♦ 1 ✱ ❡♥tã♦ 1 ✳ ▲♦❣♦✱ ϕ é s♦❜r❡❥❡✲ t♦r❛✳

      ❈❛♣ít✉❧♦ ✸ L K

      (E)

    ❝♦♠♦ ♣r♦❞✉t♦

    ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦✐❞❡ K (E)

      ◆♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r ♠♦str❛♠♦s q✉❡ ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt L é ✐s♦♠♦r❢❛ ❛ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ 1

      ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r E ✳ ◆❡st❡ ❝❛♣ít✉❧♦✱ ♠♦str❛r❡♠♦s q✉❡ ♦ ❝♦♥❥✉♥t♦ W

      ❞♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❞❡ ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ E ❛❞♠✐t❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ (E)

      ❣r✉♣♦✐❞❡ ❡ q✉❡ L K é ✐s♦♠♦r❢♦ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡st❡ ❣r✉♣♦✐❞❡✳

      ✸✳✶ ●r✉♣♦✐❞❡s

      ❉❡✜♥✐çã♦ ✸✳✶✳✶✳ ❯♠ ❣r✉♣♦✐❞❡ é ✉♠ ❝♦♥❥✉♥t♦ G ♥ã♦ ✈❛③✐♦ ❡q✉✐♣❛❞♦ ❝♦♠ ✉♠❛ ♦♣❡r❛çã♦ ❜✐♥ár✐❛ ♣❛r❝✐❛❧♠❡♥t❡ ❞❡✜♥✐❞❛ t❛❧ q✉❡✿

      ✭✐✮ ♣❛r❛ q✉❛✐sq✉❡r g, h, l ∈ G✱ g(hl) ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡ (gh)l ❡①✐st❡✳ ❊♠ ❝❛s♦ ❛✜r♠❛t✐✈♦✱ g(hl) = (gh)l❀

      ✭✐✐✮ ♣❛r❛ q✉❛✐sq✉❡r g, h, l ∈ G✱ g(hl) ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ gh ❡ hl ❡①✐st❡♠❀

      ✭✐✐✐✮ ♣❛r❛ q✉❛❧q✉❡r g ∈ G✱ ❡①✐st❡♠ ú♥✐❝♦s d(g), ǫ(g) ∈ G t❛✐s q✉❡ gd(g) ❡ ǫ(g)g ❡①✐st❡♠ ❡ gd(g) = g = ǫ(g)g❀ −1 −1

      ∈ G g ✭✐✈✮ ♣❛r❛ ❝❛❞❛ g ∈ G ❡①✐st❡ g t❛❧ q✉❡ d(g) = g ❡ ǫ(g) = −1 gg

      ✳

      ❊①❡♠♣❧♦ ✸✳✶✳✷✳ ❚♦❞♦ ❣r✉♣♦ é ✉♠ ❣r✉♣♦✐❞❡✳ ❊①❡♠♣❧♦ ✸✳✶✳✸✳ ❙❡❥❛ Gl(n, R) ♦ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s ❞❡ ♦r❞❡♠ n ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s ❡ s❡❥❛ GL(R) ♦ ❝♦♥❥✉♥t♦ ❞❛s ♠❛tr✐③❡s ✐♥✈❡rsí✈❡✐s ❝♦♠ ❝♦❡✜❝✐❡♥t❡s r❡❛✐s✳ ❈♦♥s✐❞❡r❡ ❡♠ GL(R) ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ✉s✉❛❧ ❞❡ ♠❛tr✐③❡s✳ ❊♥tã♦✱ ♣❛r❛ A, B ∈ GL(R)✱ A · B ❡stá ❞❡✜♥✐❞❛ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ A ❡ B tê♠ ✉♠❛ ♠❡s♠❛ ♦r❞❡♠ n ❡ A · B t❡♠ ♦r❞❡♠ n✳ ❆❧é♠ ❞✐ss♦✱ ❞❛❞❛ ✉♠❛ ♠❛tr✐③ A ∈ GL(R) ❞❡ ♦r❞❡♠ n ❡①✐st❡ ✉♠❛ ú♥✐❝❛ −1 −1 −1

      ∈ GL(R) = A · A = I ♠❛tr✐③ A ❞❡ ♦r❞❡♠ n t❛❧ q✉❡ A · A n ✱ ❡♠ n q✉❡ I ❞❡♥♦t❛ ❛ ♠❛tr✐③ ✐❞❡♥t✐❞❛❞❡ ❞❡ ♦r❞❡♠ n✳

      ▲♦❣♦✱ GL(R) é ✉♠ ❣r✉♣♦✐❞❡✳ Pr♦♣♦s✐çã♦ ✸✳✶✳✹✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✐❞❡✳ ❙ã♦ ✈á❧✐❞❛s ❛s s❡❣✉✐♥t❡s ❛✜r✲ ♠❛çõ❡s✿

      ✭✶✮ ❙❡ gh ❡①✐st❡✱ ❡♥tã♦ d(gh) = d(h) ❡ ǫ(gh) = ǫ(g)❀ −1 )

      ✭✷✮ d(g) = ǫ(g ✱ ♣❛r❛ t♦❞♦ g ∈ G❀ −1 ✭✸✮ g é ú♥✐❝♦✱ ♣❛r❛ t♦❞♦ g ∈ G❀ −1 −1

      ) = g ✭✹✮ (g ✱ ♣❛r❛ t♦❞♦ g ∈ G❀ −1 −1 g ✭✺✮ gh ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d(g) = ǫ(h) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ h

      ❡①✐st❡❀ −1 −1 −1 = h g

      ✭✻✮ s❡ gh ❡①✐st❡✱ ❡♥tã♦ (gh) ✳ ❉❡♠♦♥str❛çã♦✿ ✭✶✮ (gh)d(h) ❡①✐st❡ ♣♦✐s gh ❡ hd(h) ❡①✐st❡♠✳ ❉❛í✱ s❡❣✉❡ q✉❡

      (gh)d(h) = g(hd(h)) = gh.

      ▲♦❣♦✱d(h) = d(gh). ❉♦ ♠❡s♠♦ ♠♦❞♦✱ ǫ(g)(gh) ❡①✐st❡ ♣♦✐s ǫ(g)g ❡ gh ❡①✐st❡♠✳ ❉❛í✱ s❡❣✉❡ q✉❡

      ǫ(g)(gh) = (ǫ(g)g)h = gh. −1 −1 ▲♦❣♦✱ ǫ(g) = ǫ(gh).

      )(gg ) ✭✷✮ Pr✐♠❡✐r❛♠❡♥t❡✱ ♥♦t❡ q✉❡ (gg ❡①✐st❡ ❡ q✉❡ −1 −1 −1 −1 −1 −1 (gg )(gg ) = g(g g)g = (gd(g))g = gg . −1 −1 −1 −1 −1

      = d(gg ) = d(gg ) = d(g ) ▲♦❣♦✱ gg ✳ ❆ss✐♠✱ ǫ(g) = gg ✱ −1

      ) ♣❡❧♦ ✐t❡♠ ❛♥t❡r✐♦r✳ ❉♦♥❞❡ ǫ(g) = d(g ✳ ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡ −1 −1 −1 −1 −1 −1

      (g g)(g −1 −1 −1 −1 −1

      g) = g (gg )g = g ǫ(g)g = g g. g = ǫ(g

      g) g = ǫ(g

      g) = ǫ(g ) ❖✉ s❡❥❛✱ g ✳ ▲♦❣♦✱ d(g) = g ✱ −1

      ) ♣❡❧♦ ✐t❡♠ ❛♥t❡r✐♦r✳ P♦rt❛♥t♦✱ d(g) = ǫ(g ✳

      , g ∈ G ✭✸✮ ❙❡❥❛ g ∈ G ❡ s❡❥❛♠ g 1 2 ❝♦♠♦ ♥♦ ✐t❡♠ (iv) ❞❛ ❞❡✜♥✐çã♦ ❞❡

      ❣r✉♣♦✐❞❡✳ ❊♥tã♦✱ g = g d(g ) = g ǫ(g) = g (gg ) = (g g)g = d(g)g = r(g )g = g . 1 1 1 −1 −1 −1 −1 1 1 2 1 2 2 2 2 2 = ǫ(g) = d(g ) g = d(g) = ǫ(g )

      ✭✹✮ ◆♦t❡ q✉❡ gg ❡ g ✳ ▲♦❣♦ −1 −1 (g ) = g

      ✱ ♣❡❧❛ ✉♥✐❝✐❞❛❞❡ ♠♦str❛❞❛ ❡♠ (3)✳ ✭✺✮ ❙✉♣♦♥❤❛ q✉❡ gh ❡①✐st❡✳ ❱❛♠♦s ♠♦str❛r q✉❡ d(g) = ǫ(h)✳ ❈♦♠♦ gh = (gd(g))h = (gd(g))(ǫ(h)h) = g(d(g)ǫ(h))h,

      ❡♥tã♦ d(g)ǫ(h) ❡①✐st❡ ♣❡❧♦ ✐t❡♠ (ii) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❣r✉♣♦✐❞❡✳ ❉❛í✱ ❝♦♠♦ d(g) ❡ ǫ(h) sã♦ ✐❞❡♥t✐❞❛❞❡s✱ s❡❣✉❡ q✉❡ d(g) = d(g)ǫ(h) = ǫ(h).

      ❙✉♣♦♥❤❛ q✉❡ d(g) = ǫ(h)✳ ❊♥tã♦ gǫ(h) ❡①✐st❡✳ ❈♦♠♦ gǫ(h) = −1 −1 g(hh ) = (gh)h ✱ ❡♥tã♦ s❡❣✉❡ ♣❡❧♦ ✐t❡♠ (ii) ❞❛ ❞❡✜♥✐çã♦ ❞❡

      ❣r✉♣♦✐❞❡ q✉❡ gh ❡①✐st❡✳ −1 −1 g ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ q✉❡ gh ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ h ❡①✐st❡✳ ❙❛❜❡♠♦s q✉❡ gh ❡①✐st❡ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ d(g) = ǫ(h)✳ −1 −1 −1 −1

      ) ) ) = ǫ(g ) ❈♦♠♦ d(g) = ǫ(g ❡ ǫ(h) = d(h ✱ ❡♥tã♦ d(h s❡✱ −1 −1 g ❡ s♦♠❡♥t❡ s❡✱ h ✳

      ✭✻✮ ❇❛st❛ ♥♦t❛r q✉❡ −1 −1 −1 −1 −1 (gh)(h g ) = g(hh )g = gǫ(h)g −1 −1

      = gd(g)g = gg = ǫ(g) = ǫ(gh) ❡ t❛♠❜é♠ −1 −1 −1 −1 −1

      (h g )(gh) = h (g g)h = h d(g)h −1 −1 −1 −1 −1 = h ǫ(h)h = h h = d(h) = d(gh).

      = h g ▲♦❣♦ (gh) ✳

      ❖❜s❡r✈❛çã♦ ✸✳✶✳✺✳ ◆♦t❡ q✉❡ d(g) ❡ ǫ(g) sã♦ ✐❞❡♠♣♦t❡♥t❡s✱ ♣❛r❛ t♦❞♦ g ∈ G ✳ ❉❡ ❢❛t♦✱ −1 −1 −1 2 d(g) = g g = g (gd(g)) = (g g)d(g) = d(g)

      ❡ −1 −1 −1 2 ǫ(g) = gg = (ǫ(g)g)g = ǫ(g)(gg ) = ǫ(g) .

      ❉❡✜♥✐çã♦ ✸✳✶✳✻✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✐❞❡✳ ❖ ❝♦♥❥✉♥t♦ ❞♦s ♣❛r❡s ❛❞♠✐ssí✲ 2 = {(g, h) ∈ G × G : d(g) = ǫ(h)}

      ✈❡✐s ❞❡ ● é ♦ ❝♦♥❥✉♥t♦ G ✳ ❉❡✜♥✐çã♦ ✸✳✶✳✼✳ ❯♠ ❡❧❡♠❡♥t♦ e ∈ G é ❝❤❛♠❛❞♦ ❞❡ ✐❞❡♥t✐❞❛❞❡ ❞❡ ● −1

      ) s❡ e = d(g) = ǫ(g ✱ ♣❛r❛ ❛❧❣✉♠ g ∈ G✳ ❉❡♥♦t❛♠♦s ♣♦r G ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s ✐❞❡♥t✐❞❛❞❡s ❞❡ G✳ −1

      = {g g : g ∈ G} ❖❜s❡r✈❛çã♦ ✸✳✶✳✽✳ ◆♦t❡ q✉❡ G ✳ −1

      = e Pr♦♣♦s✐çã♦ ✸✳✶✳✾✳ P❛r❛ t♦❞♦ e ∈ G ✱ t❡♠♦s q✉❡ e ❡ d(e) = ǫ(e) = e

      ✳ 2 2 = d(g) = e

      ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ d(g) ✱ ❡♥tã♦ e ✳ ❉❛í d(e) = ǫ(e) = −1 e = e ✳ ❆❧é♠ ❞✐ss♦✱ ❝♦♠♦ ee = e✱ ❡♥tã♦ e ✳ 1 , · 1 ), (G 2 , · 2 ) 1 ⊔ G 2

      ❊①❡♠♣❧♦ ✸✳✶✳✶✵✳ ❙❡❥❛♠ (G ❣r✉♣♦s✳ ❙❡❥❛ G = G 2 = {(g, h) ∈ G × G : g, h ∈ G 1 2 }

      ✭✉♥✐ã♦ ❞✐s❥✉♥t❛✮ ❡ s❡❥❛ G ♦✉ g, h ∈ G ✳ ❈♦♥s✐❞❡r❡ ❛ s❡❣✉✐♥t❡ ♦♣❡r❛çã♦ ❡♠ G✿ gh = g · 1 s❡ g, h ∈ G h, h . 1 ♦✉ gh = g · 2 s❡ g, h ∈ G 2

      ❊♥tã♦ G é ✉♠ ❣r✉♣♦✐❞❡✳ ❊①❡♠♣❧♦ ✸✳✶✳✶✶✳ ❙❡❥❛ X ✉♠ ❝♦♥❥✉♥t♦ ♥ã♦✲✈❛③✐♦ q✉❛❧q✉❡r ❡ s❡❥❛ R ⊆ X × X

      ✉♠❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ 2 ′ ′ ❱❛♠♦s ♠♦str❛r q✉❡ R ❛❞♠✐t❡ ✉♠❛ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦✐❞❡✱ s❡♥❞♦ R = {((x, y), (y , z)) ∈ R × R : y = y }

      ❡ ❞❡✜♥✐♥❞♦ (x, y) · (y, z) = (x, z), 2

      ♣❛r❛ t♦❞♦ ((x, y), (y, z)) ∈ R ✱ ❡ −1 (x, y) = (y, x),

      ♣❛r❛ t♦❞♦ (x, y) ∈ R✳ Pr✐♠❡✐r❛♠❡♥t❡✱ ✈❡❥❛ q✉❡ ❛ ♦♣❡r❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❡❧❛ tr❛♥s✐✲ t✐✈✐❞❛❞❡ ❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛ ❡ ♦ ✐♥✈❡rs♦ ❡stá ❜❡♠ ❞❡✜♥✐❞♦ ♣❡❧❛

      ♣r♦♣r✐❡❞❛❞❡ s✐♠étr✐❝❛ ❞❛ r❡❧❛çã♦ ❞❡ ❡q✉✐✈❛❧ê♥❝✐❛✳ ✭✐✮ ❙❡❥❛♠ (a, b), (x, y), (g, h) ∈ R✳ ❙✉♣♦♥❤❛ q✉❡ 2 (((a, b) · (x, y)), (g, h)) ∈ R . 2

      ❊♥tã♦ g = y ❡ b = x✳ ❈♦♠♦ b = x✱ ❡♥tã♦ ((a, b), (x, y)) ∈ R ❡ 2 (a, b) · (x, y) = (a, y)

      ✳ ❈♦♠♦ y = g✱ ❡♥tã♦ ((a, y), (g, h)) ∈ R ❡ (a, y) · (g, h) = (a, h)

      ✳

      2

      ▲♦❣♦✱ (((a, b) · (x, y)), (g, h)) ∈ R ✳ 2 ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ (((a, b) · (x, y)), (g, h)) ∈ R ✳ 2

      ❊♥tã♦ b = x ❡ y = g✳ ❈♦♠♦ b = x✱ ❡♥tã♦ ((a, b), (x, y)) ∈ R ❡ 2 (a, b) · (x, y) = (a, y)

      ✳ ❈♦♠♦ y = g✱ ❡♥tã♦ ((a, y), (g, h)) ∈ R ❡ (a, y) · (g, h) = (a, h)

      ✳ 2 ▲♦❣♦✱ ((a, b), ((x, y), (g, h))) ∈ R ✳ ❊✱ (a, b)·((x, y)·(g, h)) = (a, h) = (a, b)·(x, h) = (a, b)·((x, y)·(g, h)). 2

      ✭✐✐✮ ❱❡❥❛ q✉❡✱ ❞♦ ✐t❡♠ ❛♥t❡r✐♦r✱ t❡♠♦s q✉❡ s❡ (a, b)·((x, y)·(g, h)) ∈ R 2 ❡♥tã♦ g = y ❡ b = x✳ ▲♦❣♦✱ ((a, b), (x, y)) ∈ R ❡ ((x, y), (g, h)) ∈ 2 R

      ✳ 2 2 ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ ((a, b), (x, y)) ∈ R ❡ ((x, y), (g, h)) ∈ R ✳ ❊♥tã♦ b = x ❡ y = g✳ ▲♦❣♦✱ (x, y) · (g, h) = (x, h) ❡ ❡♥tã♦ 2

      ((a, b), (x, h)) ∈ R ✳

      ✭✐✐✐✮ ❉❛❞♦ (x, y) ∈ R✱ ❝♦♥s✐❞❡r❡ (x, x), (y, y) ∈ R✳ 2 ➱ ❝❧❛r♦ q✉❡ ((x, y), (y, y)), ((x, x), (x, y)) ∈ R ❡ t❛♠❜é♠ (x, y) · (y, y) = (x, y) = (x, x)(x, y).

      ▲♦❣♦✱ (y, y) = d(x, y) ❡ (x, x) = ǫ(x, y)✳ −1 = (y, x) ∈ R

      ✭✐✈✮ ❙❡❥❛ (x, y) ∈ R✳ ❉❛í (x, y) ❡ t❡♠♦s q✉❡ ( (x, y) · (y, x) = (x, x) = ǫ(x, y) (y, x) · (x, y) = (y, y) = d(x, y) 1

      , E , r, s) ❊①❡♠♣❧♦ ✸✳✶✳✶✷✳ ❙❡❥❛ E = (E ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❡ s❡❥❛ W ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❞❡ E✳ 1 ∗ 1

      ) ❙❡❥❛ (E ✉♠ ❝♦♥❥✉♥t♦ ❞✐s❥✉♥t♦ ❞❡ E ❡ ❞❡ ♠❡s♠❛ ❝❛r❞✐♥❛❧✐❞❛❞❡✳

      ❋✐①❡ ✉♠❛ ❜✐❥❡çã♦ e 7→ e ✳ ❆❣♦r❛ ❡st❡♥❞❡♠♦s ❛s ❛♣❧✐❝❛çã♦ r, s ♣❛r❛ 1 ∗ ∗ ∗ 1 (E ) ) = r(e) ) = s(e)

      ♣♦r s(e ❡ r(e ✱ ♣❛r❛ t♦❞♦ e ∈ E ✳ 1 . . . α n ∈ W := α . . . α ∗ ∗ ∗ P❛r❛ α = α ✱ ❞❡✜♥❛ α n 1 1 ∗ 1 ✳ 1 n i i i +1 . . . α : α ∈ E ∪ (E ) ∪ E ) = s(α )}

      ❙❡❥❛ P = {α ❡ r(α ✳ e e e P♦r ❡①❡♠♣❧♦✱ ✉♠ ❡❧❡♠❡♥t♦ ❞❛ ❢♦r♠❛ e 1 2 3 4 ✱ q✉❡ ♣♦❞❡ s❡r r❡♣r❡✲ s❡♥t❛❞♦ ❝♦♠♦ ❛❜❛✐①♦✱ e e 1 2

      // • // • • e 3 OO e 4 // • •

    • e
    • 1 // • • e 2 oo e 3<
    • e
    • 4 // • é ✐rr❡❞✉tí✈❡❧✳

        ❱❛♠♦s ♠♦str❛r q✉❡ G ❛❞♠✐t❡ ❡str✉t✉r❛ ❞❡ ❣r✉♣♦✐❞❡✱ s❡♥❞♦ G 2 =

        ❙✉♣♦♥❤❛ q✉❡ ((gh), l) ∈ G 2 ✳ ❊♥tã♦ (g, h) ∈ G 2 ❡ r(g) = s(h) ❡ ❝♦♠♦ ((gh), l) ∈ G 2 ❡♥tã♦ r(h) = r(gh) = s(l)✳ ▲♦❣♦✱ (h, l) ∈ G 2

        (g, h) ∈ G 2 ✳ ❈♦♠♦ r(gh) = r(h) = s(l)✱ ❡♥tã♦ ((gh), l) ∈ G 2

        ✭✐✮ ❙✉♣♦♥❤❛ q✉❡ (g, (hl)) ∈ G 2 ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ (h, l) ∈ G 2 ❡ r(h) = s(l) ✳ ❈♦♠♦ (g, (hl)) ∈ G 2 ✱ ❡♥tã♦ r(g) = s(hl) = s(h)✳ ▲♦❣♦

        α · β = irr(αβ) ✳

        → G ❞❛❞❛ ♣♦r

        {(α, β) ∈ G × G : r(α) = s(β)} ❡ ❛ ♦♣❡r❛çã♦ ♣❛r❝✐❛❧ G 2

        α = α 1 . . . α n ∈ G ✳

        ♣❡rt❡♥❝❡ ❛ P ✳ ❉❡✜♥❛ ❡♠ P ✉♠❛ ♦♣❡r❛çã♦ · ❞❡ ❝♦♥❝❛t❡♥❛çã♦ ❞❡ ❢♦r♠❛ q✉❡ e 1 · e 2 = e 1 e 2

        ) ✱ ♣❛r❛ t♦❞♦

        ❙❡❥❛ G ♦ ❝♦♥❥✉♥t♦ ❞❛s s❡q✉ê♥❝✐❛s ✐rr❡❞✉tí✈❡✐s ❞❡ P ✳ ◆♦t❡ q✉❡ t♦❞❛ s❡q✉ê♥❝✐❛ α ❞❡ P ♣♦❞❡ s❡r r❡❞✉③✐❞❛ ❛ ✉♠❛ s❡q✉ê♥❝✐❛ irr(α) ∈ G✳ ❊s✲ t❡♥❞❛ ❛s ❛♣❧✐❝❛çõ❡s r, s ♣♦r r(α) = r(α n ) ❡ s(α) = s(α 1

        ❡ a 1 · · · a k r(a k )a k +1 · · · a n = a 1 · · · a k s(a k +1 )a k +1 · · · a n q✉❡ ♣❡rt❡♥ç❛♠ ❛ P ✱ ❞✐③❡♠♦s q✉❡ a 1 · · · a k a k +1 a n é ✉♠❛ r❡❞✉çã♦ ❞❡st❛s s❡q✉ê♥❝✐❛s✳ ❯♠❛ s❡q✉ê♥❝✐❛ ❞❡ P é ❞✐t❛ ✐rr❡❞✉tí✈❡❧ s❡ ♥ã♦ ❛❞♠✐t❡ r❡❞✉çã♦✳ P♦r ❡①❡♠♣❧♦✱

        , . . . , a n , b ∈ E 1 ∪ (E 1 ) ∪ E ❡ s❡q✉ê♥❝✐❛s ❞❛s ❢♦r♠❛s a 1 · · · a k bb a k +1 a n

        ❡ ❡st❛ ♦♣❡r❛çã♦ é ❡st❡♥❞✐❞❛ ♣❛r❛ ❝❛♠✐♥❤♦s ❞❛ ♠❛♥❡✐r❛ ♥❛t✉r❛❧✳ ❉❛❞♦s a 1

        ❀ e · e = r(e) ❀

        ✭❝♦♥❝❛t❡♥❛çã♦✮❀ e · r(e) = s(e) · e = e ❀ e · e = s(e)

        ❈♦♠♦ r(g) = s(h) = s(hl)✱ ❡♥tã♦ (g, (hl)) ∈ G 2

        2 2

        ✭✐✐✮ ❙✉♣♦♥❤❛ q✉❡ (g, (hl)) ∈ G ✳ ❊♥tã♦ (h, l) ∈ G ✳ ❊✱ ❝♦♠♦ (g, (hl)) ∈ 2 2 G ✱ ❡♥tã♦ r(g) = s(hl) = s(h)✳ ▲♦❣♦ (g, h) ∈ G ✳ 2

        ❙✉♣♦♥❤❛ q✉❡ (g, h), (h, l) ∈ G ✳ ❊♥tã♦ r(g) = s(h) ❡ r(h) = s(l)✳ 2 ❊♥tã♦ r(g) = s(h) = s(hl) ❡✱ ♣♦rt❛♥t♦✱ (g, (hl)) ∈ G ✳ 2

        ✭✐✐✐✮ ❙❡❥❛ α ∈ G✳ ◆♦t❡ q✉❡ (s(α), α) ∈ G ✱ ♣♦✐s s(α) = r(s(α)) ❡ 2 (α, r(α)) ∈ G

        ✱ ♣♦✐s r(α) = s(r(α)) = r(α)✳ ❊♥tã♦ s(α) = ǫ(α) ❡ r(α) = d(α) ✳ ∗ ∗ 2

        ), (α , α) ∈ G ✭✐✈✮ ❉❛❞♦ α ∈ G✱ ♥♦t❡ q✉❡ (α, α ✳ ❆❧é♠ ❞✐ss♦✱ ∗ ∗ αα = s(α) α = r(α).

        ❡ α ❉❡✜♥✐çã♦ ✸✳✶✳✶✸✳ ❖ ❣r✉♣♦✐❞❡ G ❞❡✜♥✐❞♦ ❛❝✐♠❛ s❡rá ❝❤❛♠❛❞♦ ❞❡ ❣r✉✲ ♣♦✐❞❡ ❞♦s ❝❛♠✐♥❤♦s✳ ❖❜s❡r✈❛çã♦ ✸✳✶✳✶✹✳ ◆♦t❡ q✉❡ ♣♦❞❡♠♦s ✈❡r W ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ G

        ✳

        ✸✳✷ ❆çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡

        ◆❡st❛ s❡çã♦ ✐♥tr♦❞✉③✐r❡♠♦s ♦ ❝♦♥❝❡✐t♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉✲ ♣♦✐❞❡ s♦❜r❡ ✉♠ ❛♥❡❧ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✽❪ ❡ ❬✾❪✳ ❉❡✜♥✐çã♦ ✸✳✷✳✶✳ ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ α ❞❡ ✉♠ ❣r✉♣♦✐❞❡ G ❡♠ ✉♠ ❝♦♥✲ g } g , {α g } g ) ❥✉♥t♦ X é ✉♠ ♣❛r α = ({X ∈G ∈G t❛❧ q✉❡✿ ǫ g

        ✭■✮ ♣❛r❛ t♦❞♦ g ∈ G✱ X (g) é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X ❡ X é ✉♠ s✉❜✲ ǫ ❝♦♥❥✉♥t♦ ❞❡ X (g) ❀ g : X −1 → X g

        ✭■■✮ α g é ✉♠❛ ❜✐❥❡çã♦ q✉❡ s❛t✐s❢❛③✿ e e ✭✐✮ α é ❛ ✐❞❡♥t✐❞❛❞❡ Id −1 X ❞❡ X ✱ ♣❛r❛ t♦❞♦ e ∈ G ❀ e 2

        (X −1 ∩ X h ) ⊆ X −1 ✭✐✐✮ α g ✱ s❡♠♣r❡ q✉❡ (g, h) ∈ G ❀ g (α h (x)) = α gh (x) (X g ∩ X h ) h (gh) −1 −1

        ✭✐✐✐✮ α ✱ ♣❛r❛ t♦❞♦ x ∈ α h ❡ 2 (g, h) ∈ G

        ✳ ❉❡✜♥✐çã♦ ✸✳✷✳✷✳ ❯♠❛ ❛çã♦ ♣❛r❝✐❛❧ α ❞❡ ✉♠ ❣r✉♣♦✐❞❡ G ❡♠ ✉♠ ❛♥❡❧ R g } g , {α g } g )

        é ✉♠ ♣❛r α = ({D ∈G ∈G t❛❧ q✉❡✿ ǫ g ǫ ✭■✮ ♣❛r❛ t♦❞♦ g ∈ G✱ D (g) é ✉♠ ✐❞❡❛❧ ❞❡ R ❡ D é ✉♠ ✐❞❡❛❧ ❞❡ D (g) ❀ g : D −1 → D g

        ✭■■✮ α g é ✉♠ ✐s♦♠♦r✜s♠♦ q✉❡ s❛t✐s❢❛③✿

        ✭✐✮ α e é ❛ ✐❞❡♥t✐❞❛❞❡ Id D e ❞❡ D e ✱ ♣❛r❛ t♦❞♦ e ∈ G ❀ ✭✐✐✮ α −1 h

        ✭✐✐✮ α −1 g = α g −1

        = α −1 h (D d (g) ∩ D ǫ (h) ) = α −1 h (D ǫ (h) ) = α −1 h (D h ) = D h −1 = D d (h) = D d (gh) = D ǫ ((gh) −1 ) = D (gh) −1 = dom(α gh ).

        ❡♥tã♦ d(g) = ǫ(h)✳ ❉❛í✱ ♣❛r❛ q✉❛✐sq✉❡r (g, h) ∈ G 2 t❡♠♦s dom(α g α h ) = α −1 h (D g −1 ∩ D h ) = α −1 h (D d (g) ∩ D ǫ (h) )

        D ǫ (g) = D gg −1 = im(α g α g −1 ) = α g (D g −1 ∩ D g −1 ) = D g . ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ D g = D ǫ (g) ✱ ♣❛r❛ t♦❞♦ g ∈ G✳ ❊♥tã♦ D g = D ǫ (g) = D d (g −1 ) ❡ ❧❡♠❜r❡ q✉❡✱ s❡ (g, h) ∈ G 2

        ♣❛r❛ t♦❞♦ (g, h) ∈ G 2 ✳ ❊♠ ♣❛rt✐❝✉❧❛r✱ t❡♠♦s✱ ♣❛r❛ t♦❞♦ g ∈ G✱

        ❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛ α ❣❧♦❜❛❧✳ ❊♥tã♦ α g (D g −1 ∩ D h ) = im(α g α h ) = im(α gh ) = D gh ,

        ✱ ♣❛r❛ t♦❞♦ g ∈ G❀ ✭✐✐✐✮ α h (D h −1 ∩ D g ) = D h ∩ D hg ✱ ♣❛r❛ t♦❞♦ (h, g) ∈ G 2

        ❣r✉♣♦✐❞❡ G ❡♠ ✉♠ ❛♥❡❧ R✳ ❊♥tã♦ ❛s s❡❣✉✐♥t❡s ❛✜r♠❛çõ❡s sã♦ ✈á❧✐❞❛s✿ ✭✐✮ α é ❣❧♦❜❛❧ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ D g = D ǫ (g) ✱ ♣❛r❛ t♦❞♦ g ∈ G❀

        (D g −1 ∩ D h ) ⊆ D (gh) −1 ✱ s❡♠♣r❡ q✉❡ (g, h) ∈ G 2

        , {α g } g ∈G ) ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠

        ✳ ▲❡♠❛ ✸✳✷✳✺✳ ❙❡❥❛ α = ({D g } g ∈G

        ✳ ❖❜s❡r✈❛çã♦ ✸✳✷✳✸✳ ◆♦t❡ q✉❡ ♦ ❞♦♠í♥✐♦ ❞❡ α g α h é α h −1 (D g −1 ∩ D h ) ❡ ❛ ✐♠❛❣❡♠ ❞❡ α g α h é α g (D g −1 ∩ D h ) ✳ ❉❡✜♥✐çã♦ ✸✳✷✳✹✳ ❉✐③❡♠♦s q✉❡ α é ❣❧♦❜❛❧ s❡ α g α h = α gh ✱ ♣❛r❛ t♦❞♦ (g, h) ∈ G 2

        ❡ (g, h) ∈ G 2

        ✱ ♣❛r❛ t♦❞♦ x ∈ α −1 h (D g −1 ∩ D h )

        ✭✐✐✐✮ α g (α h (x)) = α gh (x)

        ✭✐✐✮ ❙❡❣✉❡ ❞✐r❡t❛♠❡♥t❡ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡✳

        ✭✐✐✐✮ P❡❧♦ ✐t❡♠ (ii) ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡ ❝♦♠ g = −1 (hg) −1 −1 −1 −1 −1 −1 −1 −1 −1 ❡ ♥♦t❛♥❞♦ q✉❡

        ((hg)

        h) = (g d(h)) = (g d(g )) = (g ) = g, −1 (D hg ∩ D h ) ⊆ D −1 −1 = D g t❡♠♦s q✉❡ α h ✳ h −1 h hg ∩ D h ⊆ h ((hg) )

        ❈♦♠♦ α é ✐s♦♠♦r✜s♠♦ ❞❡ D h ❡♠ D ✱ t❡♠♦s q✉❡ D α h (D g ) = α h (D −1 ∩ D g ) h (D g ∩ D h ) ⊆ D h h ✳ −1 ➱ ❝❧❛r♦ q✉❡ α ✳ P❡❧♦ ✐t❡♠ (ii) ❞❛ ❞❡✜♥✐çã♦ −1 −1 ❞❡ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡✱ ❝♦♠ g = g ❡ h = h ✱ t❡♠♦s −1 −1 α h (D g ∩ D h ) ⊆ D hg h (D g ∩ D h ) ⊆ D h ∩ D hg

        ✳ ▲♦❣♦✱ α ✳

        

      ✸✳✸ ➪❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❝♦♠♦ ♣r♦❞✉t♦ ♣❛r✲

      ❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡

        ◆❡st❛ s❡çã♦✱ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❡ ✉♠❛ ❛çã♦ ❞❡ ✉♠ ❣r✉♣♦✐❞❡ ❞❡ ❛❝♦r❞♦ ❝♦♠ ❬✽❪ ❡ ❬✾❪✳ ❊♠ s❡❣✉✐❞❛✱ ❞❡✜♥✐r❡♠♦s ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ❞♦s ❝❛♠✐♥❤♦s ❡ ♠♦str❛r❡♠♦s q✉❡ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ❡ss❛ ❛çã♦ é ✐s♦♠♦r❢♦ à á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt✳

        } , {α } ) ❙❡❥❛ α = ({D g g ∈G g g ∈G ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ❡♠ α G

        ✉♠ ❛♥❡❧ R✳ ❙❡❥❛ R ⋊ ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦♠❛s ❢♦r♠❛✐s ✜♥✐t❛s X a g δ g g ∈ D g ❞❛ ❢♦r♠❛ ✱ ❝♦♠ a ✳ g ∈G α G

        ❈♦♥s✐❞❡r❡ ❡♠ R ⋊ ✱ ❛ ❛❞✐❝ã♦ ✉s✉❛❧ ❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❞❛❞❛ ♣♦r ( −1 2 α g (α g (a g )b h )δ gh ;

        ✱ s❡ (g, h) ∈ G a g δ g · b h δ h = ✱ ❝❛s♦ ❝♦♥trár✐♦✳

        ❱❡❥❛ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s✱ ❝♦♠♦ d(g) = ǫ(h)✱ 2 −1 h = ♣❛r❛ t♦❞♦ (g, h) ∈ G ✱ ❡♥tã♦ D g ❡ D sã♦ ❛♠❜♦s ✐❞❡✐❛s ❞❡ D d (g) −1 D (a g )b h ∈ D −1 ∩ D h ǫ (h) ✳ ▲♦❣♦✱ α g g ✳ ❆❧é♠ ❞✐ss♦✱ ♣❡❧♦ ✐t❡♠ (iii) ❞❡ s❡❣✉❡ q✉❡ 2 α g (D −1 ∩ D h ) = D g ∩ D gh , g g (α −1 (a g )b h ) ∈ D gh

        ♣❛r❛ t♦❞♦ (g, h) ∈ G ✳ ▲♦❣♦✱ α g ✳ α G ❉❡✜♥✐çã♦ ✸✳✸✳✶✳ R ⋊ ❝♦♠ ❛s ♦♣❡r❛çõ❡s ❛❝✐♠❛ é ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ❛ss♦❝✐❛❞♦ à α✳

        ❙❡❥❛ E = (E , E 1 , r, s)

        θ b −1 (ξ) = ( ξ |b|+1 ξ |b|+2 . . .

        ) = gξ = ξ ❡✱ ♣♦rt❛♥t♦✱

        ) ✳ ❊♥tã♦ θ g (ξ

        é ❜✐❥❡çã♦ ✿ ➱ ❝❧❛r♦ q✉❡ θ g é ✐♥❥❡t✐✈❛✳ ❆❣♦r❛✱ ❞❛❞♦ ξ ∈ X g ✱ ❡♥tã♦ ξ = gξ ✳ ▲♦❣♦✱ s(ξ) = s(g) = r(g −1

        ✭■■✮ θ g : X g −1 → X g

        ✳ Pr♦♣♦s✐çã♦ ✸✳✸✳✷✳ θ = ({X g } g ∈G , {θ g } g ∈G ) é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ♥♦ ❝♦♥❥✉♥t♦ X✳ ❉❡♠♦♥str❛çã♦✿ ✭■✮ ➱ ❢á❝✐❧ ✈❡r q✉❡ X ǫ (g) é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X ❡ q✉❡ X g é ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡ X ǫ (g) ✳

        ξ 7→ aξ |b|+1 ξ |b|+2 . . .

        ξ 7→ bξ |a|+1 ξ |a|+2 . . . θ ab −1 : X ba −1 → X ab −1

        ✱ ❞❡✜♥❛ θ ba −1 : X ab −1 → X ba −1

        P❛r❛ ab −1 ∈ G

        ✱ s❡ r(b) ♥ã♦ é ♣♦ç♦ r(b) ✱ s❡ r(b) é ♣♦ç♦.

        → X b −1 ✱ ♣♦r

        ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❡ s❡❥❛ G ♦ ❣r✉♣♦✐❞❡ ❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ❙❡❥❛ W ⊆ G ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❞❡ E ❡ s❡❥❛ X = {ξ ∈ W : r(ξ) é ♣♦ç♦} ∪ {v ∈ E

        ξ 7→ bξ ❊ θ b −1 : X b

        P❛r❛ b ∈ W ✱ ❞❡✜♥❛ θ b : X b −1 → X b

        θ v : X v → X v ξ 7→ ξ

        ✱ ❞❡✜♥❛ X g = ∅ ✳ ❱❛♠♦s ❞❡✜♥✐r ❛❣♦r❛ ✉♠❛ ❛çã♦ ❞❡ ❣r✉♣♦✐❞❡✳ P❛r❛ v ∈ E ✱ ❞❡✜♥❛

        ✳ P❛r❛ v ∈ E ✱ ❞❡✜♥❛ X v := {ξ ∈ X : s(ξ) = v} ✳ P❛r❛ q✉❛❧q✉❡r ♦✉tr♦ g ∈ G

        ✱ ❝♦♠ a, b ∈ W ❡ a, b /∈ E ✱ ❞❡✜♥❛ X ab −1 := X a

        P❛r❛ ab −1 ∈ G

        {ξ ∈ X : r(ξ) = r(a)} ✳

        ❡ X a −1 :=

        P❛r❛ a ∈ W \ E ✱ ❞❡✜♥❛ X a := {ξ ∈ X : ξ 1 . . . ξ |a| = a}

        : v é ♣♦ç♦} ∪ W

        θ g é s♦❜r❡❥❡t✐✈❛✳ v v

        ✭✐✮ ❙❡❥❛ v ∈ E ✳ ❱❛♠♦s ♠♦str❛r q✉❡ θ é ✐❞❡♥t✐❞❛❞❡ ❞❡ X ✳ v ❉❛❞♦ ξ ∈ X ✱ ❡♥tã♦ s(ξ) = v = r(ξ)✳ ❉❛í✱ θ v (ξ) = vξ = s(ξ)ξ = ξ. −1 −1 −1

        (X g ∩ X h ) ⊆ X ✭✐✐✮ ❱❛♠♦s ♠♦str❛r q✉❡ θ h (gh) ✳ −1 −1

        , h = cd ∈ G ❙❡❥❛♠ g = ab ✳ g −1 ∩ X h = X ba −1 ∩ X cd −1 ❈♦♠♦ X ✱ ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r três ❝❛s♦s✿ ❈❛s♦ ✶✿ b ♥ã♦ é ❝♦♠❡ç♦ ❞❡ c ❡ c ♥ã♦ é ❝♦♠❡ç♦ ❞❡ b✳ −1 ∩ X −1 = ∅ ◆❡st❡ ❝❛s♦✱ X ba cd ✳ ❈❛s♦ ✷✿ b é ❝♦♠❡ç♦ ❞❡ c ❡ ❡s❝r❡✈❡♠♦s c = bc ✳  a d ✱ s❡ b = r(a), c = r(d) ∈ E X −1 ∩ X −1 X −1 ∩X −1 = ba cd a c X −1 ∩ X , c / ∈ E ✱ s❡ b ∈ E

        X b ∩ X c −1 −1 ✱ s❡ b, c /∈ E X a ∩ X d ✱ s❡ b = r(a), c = r(d) ∈ E

        = X a −1 ∩ X c , c / ∈ E ✱ s❡ b ∈ E X c

        ✱ s❡ b, c /∈ E ❊♥tã♦✱ −1 −1  θ (X a ∩ X d ) (1) d −1 −1 −1 −1

        θ (X g ∩ X h ) = θ −1 (X −1 ∩ X c ) (2) h cd a −1 θ (X ) (3) cd −1 c

        ❊♠ (1)✱ t❡♠♦s −1 −1 −1 −1 −1 −1 (gh) = h g = dc ba = da . −1 = X −1 = X d ❊♥tã♦ X (gh) da ❡✱ ♣♦rt❛♥t♦✱ −1 −1 −1 θ d (X a ∩ X d ) ⊆ X d = X . (gh) ❊♠ ✭✷✮✱ t❡♠♦s −1 −1 −1 −1 −1 −1 −1 (gh) = h g = dc ba = dc a . −1 −1 −1

        = X dc a = X d ❊♥tã♦ X (gh) ❡✱ ♣♦rt❛♥t♦✱ θ −1 (X −1 ∩ X c ) ⊆ X −1 = X d = X −1 . dc a dc (gh)

        ❊♠ ✭✸✮✱ t❡♠♦s −1 −1 −1 ′−1 −1 −1 ′−1 −1 ′−1 −1 (gh) = dc ba = dc b ba = dc r(b)a = dc a . dc ⊆ X dc = X dc b = X . −1 −1 ′−1 −1 −1 ❊♥tã♦ θ (gh) ❈❛s♦ ✸✿ c é ❝♦♠❡ç♦ ❞❡ b ❡ ❡s❝r❡✈❡♠♦s b = cb ✳ ❊ss❡ ❝❛s♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ h ◦ θ g (ξ) = θ hg (ξ)

        ✭✐✐✐✮ ❱❛♠♦s ♠♦str❛r q✉❡ θ ✱ ♣❛r❛ t♦❞♦ ξ ∈ −1 −1 θ (X g ∩ X h ) g ✳ ❚❡♠♦s três ❝❛s♦s✳ ❈❛s♦ ✶✿ a ♥ã♦ é ❝♦♠❡ç♦ ❞❡ d ❡ d ♥ã♦ é ❝♦♠❡ç♦ ❞❡ a✳ h −1 ∩ X g = ∅ ◆❡st❡ ❝❛s♦✱ X ✳ ❈❛s♦ ✷✿ a é ❝♦♠❡ç♦ ❞❡ d ❡ ❡s❝r❡✈❡♠♦s d = ad ✳ −1

        (X g ∩ X −1 ) = θ −1 (X −1 ∩ X −1 ) ❙❡❥❛ ξ ∈ θ g h ba ab dc ✳ ❊♥tã♦ ξ = θ ba −1 (η) ab −1 ∩ X dc −1

        ✱ ♣❛r❛ ❛❧❣✉♠ η ∈ X ✳ ❉❛í✱ ′ ′ ′ ′ ξ = θ −1 (η) = θ −1 (ad ξ ) = bd η . ba ab P♦r ♦✉tr♦ ❧❛❞♦✱ ❝♦♠♦ −1 −1 ′ −1 −1 ′ −1 ′ −1 hg = cd ab = cd a ab = cd r(a)b = cd b , t❡♠♦s ′ ′ ′ θ hg (ξ) = θ ′ −1 ((bd )η ) = cη . c (bd ) ❈❛s♦ ✸✿ d é ❝♦♠❡ç♦ ❞❡ a ❡ ❡s❝r❡✈❡♠♦s a = da ✳ ❊st❡ ❝❛s♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳ g } g , {θ g } g )

        P♦rt❛♥t♦✱ θ = ({X ∈G ∈G é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡✳ ❱❛♠♦s ❞❡✜♥✐r ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡ ♥♦ ♥í✈❡❧ ❞❡ ❛♥❡❧✳ ❙❡❥❛ K ✉♠ ❝♦r♣♦ ❡ s❡❥❛ F(X) = {f : X → K|f é ❢✉♥çã♦}✳ P❛r❛

        X g 6= 0 g ) = {f ∈ F(X) : f g } ⊆ F ✱ ❝♦♥s✐❞❡r❡ F(X s❡ ❛♥✉❧❛ ❢♦r❛ ❞❡ X (X) g = ∅ g ) = {

        ❡ ♣❛r❛ X ✱ s❡❥❛ F(X ❢✉♥çã♦ ♥✉❧❛ }✳ ❉❡✜♥❛ t❛♠❜é♠ F F −1

        α g : (X g ) → (X g ) −1 f 7→ f ◦ θ g ❡ F F

        α v : (X v ) → (X v ) f 7→ f ◦ θ v g )} g , {α g } g )

        Pr♦♣♦s✐çã♦ ✸✳✸✳✸✳ α = ({F(X ∈G ∈G é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ♥♦ ❛♥❡❧ F(X)✳ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ ♦ ú♥✐❝♦ ✐t❡♠ ♥ã♦ tr✐✈✐❛❧ ❞❛ ❞❡✜♥✐çã♦ ❞❡ ❛çã♦ ♣❛r❝✐❛❧ é ♦ ✐t❡♠ (II) − (ii)✳ −1 −1 −1

        (F(X g )∩F(X h )) ) ❉❛❞❛ f ∈ α h ✱ ✈❛♠♦s ♠♦str❛r q✉❡ f ∈ F(X (gh) ✳ h ) −1 X = 0 ◆♦t❡ q✉❡✱ ❡♠ ♣❛rt✐❝✉❧❛r✱ f ∈ F(X ✱ ✐st♦ é✱ f| \X ✳ ❊♥tã♦✱ X = 0 h−1

        ❞❡✈❡♠♦s ♠♦str❛r q✉❡ f| \X ✳ (gh)−1 ❚❛♠❜é♠✱ (X \ X −1 ) \ (X \ X −1 ) = X −1 \ (X −1 ∩ X −1 ). (gh) h h (gh) h h \ (X ∩ X h ) −1 −1 −1 ❚♦♠❡ x ∈ X (gh) ✳ ❊♥tã♦ −1 −1 −1 −1 θ h (x) ∈ X h \ (θ h (X ∩ X h ) = X h \ X g , (gh)

        (X −1 ∩ X p ) ⊆ X −1 ♣♦✐s θ p q (qp) ✳

        ▲♦❣♦✱ h −1 \ (X −1 ∩ X h −1 ) 0 = α h (f )(θ h (x)) = f (x), ♣❛r❛ t♦❞♦ x ∈ X (gh) ✳ −1 ) ∪ (X −1 \ (X −1 ∩ X −1 )) =

        P♦rt❛♥t♦✱ f é ③❡r♦ ❡♠ (X \ X h h (gh) h X \ X −1 (gh) ✳ g : g ∈ G} p = 1 p D(X) = span{1 p 1 g :

        ❈♦♥s✐❞❡r❡ D(X) = span{1 ❡ D g ∈ G} p ✱ ♣❛r❛ t♦❞♦ p ∈ G ✭span ❑✲❧✐♥❡❛r✮✱ ❡ ❝♦♥s✐❞❡r❡ ❛ r❡str✐çã♦ ❞❡ α p

        ❛♦s ✐❞❡❛✐s D ✱ α p : D −1 → D p p g } g , {α g } g ) 1 −1 p p 1 q 7→ α p (1 −1 1 p ) = 1 p 1 pq

        ❊♥tã♦ ˜α = ({D ∈G ∈G é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ♥♦ ❛♥❡❧ D(X)✳ α G

        ❙❡❥❛ D(X) ⋊ ˜ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❛ss♦❝✐❛❞♦ ❛ ˜α✳ ❖❜s❡r✈❛çã♦ ✸✳✸✳✹✳ ◆♦ ❝❛♣ít✉❧♦ ✷✱ ❝♦♠ ❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ F✱ p : p ∈ F \ {0}} ∪ {1 v : v ∈ E }} ❝♦♥s✐❞❡rá✈❛♠♦s D(X) = span{{1 ✳ ❆❣♦r❛✱ ❞❡✈❡♠♦s t♦♠❛r ✉♠ ❝❡rt♦ ❝✉✐❞❛❞♦ q✉❛♥❞♦ v é ♣♦ç♦✳ ▼❛s✱ s❡ v é v = {v} v δ v ∈ D(X) ⋊ α G ♣♦ç♦✱ ❡♥tã♦ X ❡✱ ♣♦rt❛♥t♦✱ 1 ˜ ✳ K (E) α Pr♦♣♦s✐çã♦ ✸✳✸✳✺✳ ❊①✐st❡ ✉♠ K✲❤♦♠♦♠♦r✜s♠♦ φ ❞❡ L ❡♠ D(X)⋊ ˜ 1 ∗ G e δ e ) = 1 −1 δ −1 t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ e ∈ E ✱ φ(e) = 1 ❡ φ(e e e ❡✱ ♣❛r❛ v δ v t♦❞♦ v ∈ E ✱ φ(v) = 1 ✳

        ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s {1 e δ e , 1 e −1 δ e −1 : e ∈ E 1 }

        ❡ {1 v δ v : v ∈ E } ❡♠ D(X) ⋊ ˜ α G

        ❱❛♠♦s ♠♦str❛r q✉❡ t❛✐s ❝♦♥❥✉♥t♦s s❛t✐s❢❛③❡♠ ❛s r❡❧❛çõ❡s q✉❡ ❞❡✜✲ ♥❡♠ ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt✳

        ✭✐✮ 1 s (e) δ s (e) 1 e δ e = α s (e) (α s (e) −1 (1 s (e) )1 e )δ s (e)e = α s (e) (1 (s(e)) −1 1 e )δ e = 1 s (e) 1 s (e)e δ e = 1 s (e) 1 e δ e = 1 e δ e 1 e δ e

        1 r (e) δ r (e) = α e (α −1 e (1 e )1 r (e) )δ er (e) = α e (α −1 e (1 e )1 r (e) )δ e = α e (1 e −1 1 r (e))δ e = 1 e 1 er (e) δ e = 1 e δ e

        ✭✐✐✮ 1 r (e) δ r (e) 1 e −1 δ e −1 = α r (e) (α r (e) 1 (1 r (e) )1 e −1 )δ r (e)e −1 = α r (e) (1 r (e) −1 1 e −1 )δ s (e −1 )e −1 = 1 r (e) 1 r (e)e −1 δ e −1 = 1 r (e) 1 s (e −1 )e −1 δ e −1 = 1 r (e) 1 e −1 δ e −1 = 1 e −1 δ e −1

        1 e −1 δ e −1

        1 s (e) δ s (e) = α e −1 (α e (1 e −1 )1 s (e)s (e)e −1 = α e −1 (1 e

        1 s (e)e −1 r (e −1 ) = 1 e −1

        1 e −1 s (e) δ e −1 = 1 e −1

        1 e −1 δ e −1 = 1 e −1 δ e −1

        ✭✐✐✐✮ ❙❡❥❛♠ f, g ∈ E 1 ✳ ◆♦t❡ q✉❡

        1 f −1 δ f −1 1 g δ g = α f −1 (α f (1 f −1 )1 g )δ f −1 g = α f −1 (1 f 1 g )δ f −1 g . f

        1 g = 0 −1 δ −1 1 f δ f ❙❡ f 6= g✱ ♣❡❧♦ ❧❡♠❛ ✸✳✶✳✶✱ 1 ✳ ❱❡❥❛♠♦s ❛❣♦r❛ 1 f f ✳ −1 −1 −1 −1 −1 1 f δ f

        1 f δ f = α (α f (1 f )1 f )δ f f −1 f = α (1 f f (e) 1 f )δ r = α −1 (1 f )δ f r (f ) = 1 −1 δ f r (e) = 1 δ r (f ) r (f ) v =

        ✭✐✈✮ ❙❡❥❛ v ∈ E t❛❧ q✉❡ 0 &lt; #{e : s(e) = v} &lt; ∞✳ ❈♦♠♦ X [ X e X e ∈E :s(e)=v 1 ✱ ❡♥tã♦ X X e e e ∈E :s(e)=v ∈E :s(e)=v ∈E :s(e)=v 1 1 e δ e 1 e −1 δ e −1 = 1 1 e δ s = ( (e) 1 1 e )δ v = 1 v δ v .

        ⋊ F α α G Pr♦♣♦s✐çã♦ ✸✳✸✳✻✳ ❊①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ D ❡ D(X)⋊ ˜ ✳ ❉❡♠♦♥str❛çã♦✿ ❙❛❜❡♠♦s q✉❡ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦♠♦♠♦r✜s♠♦

        φ : L K (E) → D(X) ⋊ ˜ G alpha 1 δ ) = 1 −1 δ −1 δ t❛❧ q✉❡ φ(e) = 1 e e ✱ φ(e e e ✱ ♣❛r❛ t♦❞♦ e ∈ E ✱ ❡ φ(v) = 1 v v ✱

        ♣❛r❛ t♦❞♦ v ∈ E ✳ ❚❛♠❜é♠✱ ⋊ F L K (E) ∼ = D α . 1 e δ e ∈ D(X) ⋊ α G −1 δ −1 ✭⋆✮ P❛r❛ ❝❛❞❛ e ∈ E ✱ ❝♦♥s✐❞❡r❡ 1 ˜ ❡ 1 e e ✳ P❛r❛ v δ v ∈ D(X) ⋊ α G

        ❝❛❞❛ v ∈ E ✱ ❝♦♥s✐❞❡r❡ 1 ˜ ✳ (E)

        P❡❧❛ ♣r♦♣r✐❡❞❛❞❡ ✉♥✐✈❡rs❛❧ ❞❡ L K ❡ ♣♦r ⋆✱ ❡①✐st❡ ✉♠ ú♥✐❝♦ ❤♦✲ ♠♦♠♦r✜s♠♦ ⋊ F e δ e ) = 1 e δ e e δ e ) = 1 e δ e Γ : D → D(X) ⋊ G −1 −1 −1 −1 1 t❛❧ q✉❡ Γ(1 ✱ Γ(1 ✱ ♣❛r❛ t♦❞♦ e ∈ E ✱ ❡

        Γ(1 v δ ) = 1 v δ v ✱ ♣❛r❛ t♦❞♦ v ∈ E ✳ −1 −1 1

        Γ e δ e , 1 e δ e : e ∈ E } v δ v : v ∈ E } é s♦❜r❡❥❡t♦r❛ ♣♦✐s {1 ❡ {1 α G

        ❣❡r❛♠ D(X) ⋊ ✭❝♦♠♦ á❧❣❡❜r❛✮✳ ❱❛♠♦s ♠♦str❛r q✉❡ Γ é ✐♥❥❡t♦r❛✳ X

      • δ a g δ g ❙❡❥❛ x ∈ Ker Γ✳ ❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r x = a ✱ ❡♠
      • X X g ∈ D g = λ ab −1 −1 1 ab β w 1 w ∈ D q✉❡ a
      P❛r❛ 0 6= g ∈ 0✱ ❝♦♠♦ Γ é ❤♦♠♦♠♦r✜s♠♦✱ é ❢á❝✐❧ ✈❡r q✉❡ Γ(a g δ g ) = a g δ g ✳

        ◆♦t❡ ❛✐♥❞❛ q✉❡✱ ❞❛❞♦ 1 a δ ∈ D δ ✱ ❝♦♠ a ∈ W ✱ t❡♠♦s 1 a δ a · 1 a −1 δ a −1 = α a (α a −1 (1 a )1 a −1 )δ = α a (1 a −1

        = {e} ✱ s(e) = v 1 ❡ r(e) = v 2 ✱ ❡ s❡❥❛ K ✉♠ ❝♦r♣♦✳

        D v 2 = span{1 v 2 } ✱ D e = span{1 e } ❡ D e

        = span{1 v 1 } ✱

        , 1 v 2 , 1 e , 1 e ∗ } ✱ D v 1

        {v 2 } ✳ ◆♦t❡ ❛✐♥❞❛ q✉❡ D(X) = span{1 v 1

        = {v 2 } ✱ X e ∗ =

        = {e} ✱ X e = {e} ✱ X v 2

        , e} ✱ X v 1

        ✳ ◆♦t❡ q✉❡ X = {v 2

        ❊♥tã♦ ♦ ❣r✉♣♦✐❞❡ ❣❡r❛❞♦ ♣♦r E 1 é {v 1 , v 2 , e, e }

        E = {v 1 , v 2 } ✱ E 1

        1 a −1 )δ = 1 a δ . ▲♦❣♦✱

        , E 1 , r, s) ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦✱ ❡♠ q✉❡

        P♦rt❛♥t♦ Γ é ✐♥❥❡t♦r❛✳ ❊①❡♠♣❧♦ ✸✳✸✳✼✳ ❙❡❥❛ E = (E

        ⇐⇒ a = 0 ❡ a g = 0, ♣❛r❛ t♦❞♦ g.

        ❉❛í✱ Γ(x) = 0 ⇐⇒ X v ∈V 1 v a δ v + X a g δ g = 0

        = {v ∈ V : v = s(ab −1 ) ♣❛r❛ λ ab −1 ♦✉ β v 6= 0}

        = X v ∈V 1 v a δ v , ❡♠ q✉❡ V

        Γ(a δ ) = Γ(( X λ ab −1 1 ab −1 + X β w 1 w )δ ) = X λ ab −1 1 ab −1 δ s (a) + X β w 1 w δ w

        ❙❡♥❞♦ ❛ss✐♠✱ ♣❡❧♦ ♦ q✉❡ ❢♦✐ ❢❡✐t♦ ❛❝✐♠❛✱ ♣❡❧♦ ❢❛t♦ ❞❡ Γ s❡r ❤♦♠♦✲ ♠♦r✜s♠♦✱ 1 cd −1 = 1 c ❡ 1 d −1 = 1 r (d) ✱ ♣❛r❛ c, d ∈ W ✱ s❡❣✉❡ q✉❡

        Γ(1 a δ ) = Γ(1 a δ a )Γ(1 a −1 δ a −1 ) = 1 a δ a · 1 a −1 δ a −1 = α a (α a −1 (1 a )1 a −1 )δ aa −1 = α a (1 a −1 )δ s (a) = 1 a δ s (a) .

        = span{1 e } ✳

        ❚❡♠♦s D(X) ⋊ G ♦ ❝♦♥❥✉♥t♦ ❞❛s s♦♠❛s ❢♦r♠❛✐s ✜♥✐t❛s λ 1 1 v 1 δ v 1 + λ 2 1 v 2 δ v 2 + λ 3 1 e δ e + λ 4 1 e δ e ,

        ♦✉ s❡❥❛✱ D(X) ⋊ G = K1 v 1 δ v 1 L K1 v 2 δ v 2 L K1 e δ e L K1 e δ e

        ✳ ❉❡ss❡ ♠♦❞♦✱ ✐❞❡♥t✐✜❝❛♠♦s D(X) ⋊ G ❝♦♠ K 4 ♣♦r

        D(X) ⋊ G → K 4 λ 1 1 v 1 δ v 1 + λ 2 1 v 2 δ v 2 + λ 3 1 e δ e + λ 4 1 e δ e 7→ (λ 1 , λ 2 , λ 3 , λ 4 ).

        P❛r❛ ❞❡t❡r♠✐♥❛r ♦ ♣r♦❞✉t♦ ❡♠ K 4 ✈✐❛ ❡st❛ ✐❞❡♥t✐✜❝❛çã♦✱ ♥♦t❡ q✉❡✱ ✉s❛♥❞♦ ♦ ✐s♦♠♦r✜s♠♦ ❞♦ t❡♦r❡♠❛✱ L K (E) → D(X) ⋊ G ✱ z 7→ 1 z δ z ✱ ♦❜t❡♠♦s

        (1 z δ z )(1 w δ w ) = ( 1 zw δ zw , s❡ z ❡ w sã♦ ❝♦♠♣♦♥í✈❡✐s

        0, ❝❛s♦ ❝♦♥trár✐♦✳

        ❙❡♥❞♦ ❛ss✐♠✱ s❡❣✉❡ q✉❡ ♦ ♣r♦❞✉t♦ ❡♠ D(X) ⋊ G ∼ = K 4 é ❞❛❞♦ ♣♦r (λ 1 , λ 2 , λ 3 , λ 4 )(µ 1 , µ 2 , µ 3 , µ 4 ) =

        (λ 1 µ 1 + λ 3 µ 4 , λ 4 µ 3 + λ 2 µ 2 , λ 1 µ 3 + λ 3 µ 2 , λ 4 µ 1 + λ 2 µ 4 ).

        ❖ ❧❡✐t♦r ❛t❡♥t♦ ♣♦❞❡ ♥♦t❛r q✉❡ ♦ ♣r♦❞✉t♦ ❛❝✐♠❛ é s❡♠❡❧❤❛♥t❡ ❛ ✉♠ ♣r♦❞✉t♦ ♠❛tr✐❝✐❛❧✳ ▼❛✐s ❞♦ ✐ss♦✱ ❛ ❛♣❧✐❝❛çã♦ φ : K 4

        → M 2 (K) ✱

        ❞❛❞❛ ♣♦r φ(λ 1 , λ 2 , λ 3 , λ 4 ) =

        λ 1 λ 3 λ 4 λ 2

        é✱ ♥❛ ✈❡r❞❛❞❡✱ ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛s✳

        ❋✐♥❛❧♠❡♥t❡✱ ♣♦❞❡♠♦s ❝♦♥❝❧✉✐r q✉❡ L K (E) é ✐s♦♠♦r❢❛ ❛ M 2 (K)

        ✳

        ❈❛♣ít✉❧♦ ✹ ■s♦♠♦r✜s♠♦s ❡♥tr❡ á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt K (E)

        ◆♦ ❝❛♣ít✉❧♦ ✷✱ ♠♦str❛♠♦s q✉❡ ❛ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt L é ✐s♦✲ ♠♦r❢❛ ❛ ✉♠ ♣r♦❞✉t♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞♦ ❣r❛❢♦ E✳ ◆♦ ❝❛♣ít✉❧♦ ✸✱ ♠♦str❛♠♦s q✉❡ L K (E)

        é ✐s♦♠♦r❢❛ ❛ ✉♠ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ ❣❡r❛❞♦ ♣❡❧❛s ❛r❡st❛s ❞❡ E✳ ❖✉ s❡❥❛✱ ♦ ♣r♦❞✉t♦ ❝r✉✲ ③❛❞♦ ♣❛r❝✐❛❧ ❝♦♥str✉í❞♦ ♥♦ ❝❛♣ít✉❧♦ ✷ é ✐s♦♠♦r❢♦ ❛♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ❝♦♥str✉í❞♦ ♥♦ ❝❛♣ít✉❧♦ ✸✳

        ◆❡st❡ ❝❛♣ít✉❧♦ ❡st✉❞❛r❡♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssár✐❛s ♣❛r❛ q✉❡✱ ❞❛❞♦s ❞♦✐s ❣r✉♣♦✐❞❡s G 1 ❡ G 2 ❛ss♦❝✐❛❞♦s ❛ ❣r❛❢♦s E 1 ❡ E 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✱ t❡♥❤❛♠♦s ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ♣r♦❞✉t♦s ❝r✉③❛❞♦s ❛ss♦❝✐❛❞♦s ❛ ❡❧❡s✳

        ✹✳✶ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦✐❞❡s

        ❉❡✜♥✐çã♦ ✹✳✶✳✶✳ ❙❡❥❛♠ G, H ❣r✉♣♦✐❞❡s✳ ❯♠❛ ❛♣❧✐❝❛çã♦ h : G → H é 2 , g ) ∈ G ), h(g )) ∈

        ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦✐❞❡s s❡ (g 2 1 2 ✐♠♣❧✐❝❛ ❡♠ (h(g 1 2 H g ) = h(g )h(g ) ❡ h(g 1 i = (E , E , r, s) i 2 1 1 2 ✳

        ❙❡❥❛ E i i ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❡ s❡❥❛ G ❣r✉♣♦✐❞❡ ❞❡ ❝❛♠✐✲ ♥❤♦ ❛ss♦❝✐❛❞♦ ❛ E i ✱ ❝♦♠♦ ❞❡✜♥✐❞♦ ❡♠ ♣❛r❛ i ∈ {1, 2}✳ ❙❡❥❛ W i ⊆ G i i i = {ξ ∈ W i :

        ♦ ❝♦♥❥✉♥t♦ ❞♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❞❡ E ❡ s❡❥❛ X r(ξ) i = ({X g } g , {θ g } g ) é ♣♦ç♦} ∪ W ✱ ♣❛r❛ i ∈ {1, 2}✳ ❙❡❥❛ θ ∈G ∈G ❛ i i i

        ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦✐❞❡ G ♥♦ ❝♦♥❥✉♥t♦ X ✱ ❝♦♠♦ ❞❡✜♥✐❞♦ ♥♦ ❝❛♣ít✉❧♦ ❛♥t❡r✐♦r✱ ♣❛r❛ i ∈ {1, 2}✳

        ❉❡✜♥✐çã♦ ✹✳✶✳✷✳ ❉❡✜♥✐♠♦s S ⊆ G ♣♦r S = {g ∈ G : X g 6= ∅}.

        → G ❈♦♥s✐❞❡r❡ h : G 1 2 ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❡♥tr❡ ♦s ❣r✉♣♦✐❞❡s✳ ❱❛✲

        ♠♦s ♠♦str❛r ❛❧❣✉♥s ❧❡♠❛s ❛ r❡s♣❡✐t♦ ❞❡ h✳ ❚❛✐s ❧❡♠❛s s❡rã♦ ✉t✐❧✐③❛❞♦s ♣❛r❛ ♦s r❡s✉❧t❛❞♦s ❞❛ ♣ró①✐♠❛ s❡çã♦✳ ▲❡♠❛ ✹✳✶✳✸✳ P❛r❛ t♦❞♦ α ∈ W 1 ✱ t❡♠♦s q✉❡ h(s(α)) = s(h(α)) ❡ h(r(α)) = r(h(α))

        ✳ ❉❡♠♦♥str❛çã♦✿ ❈♦♠♦ h(s(α)) = h(s(α)s(α)) = h(s(α))h(s(α))✱ ❡♥✲ tã♦ h(s(α)) ∈ E ✳ 1 ❈♦♠♦ h(α) = h(s(α)α) = h(s(α))h(α)✱ ❡♥tã♦ s(h(α)) = h(s(α)). ❈♦♠♦ h(r(α)) = h(r(α)r(α)) = h(r(α))h(r(α))✱ ❡♥tã♦ h(r(α)) ∈

        E 1 ✳ ❈♦♠♦ h(α) = h(αr(α)) = h(α)h(r(α), ❡♥tã♦ r(h(α)) = h(r(α))✳ −1 −1

        ) = h(α) ▲❡♠❛ ✹✳✶✳✹✳ ❉❛❞♦ α ∈ W 1 ✱ s❡❣✉❡ q✉❡ h(α ✳ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ −1 −1 −1 h(α )h(α) = h(α α) = h(r(α)) = r(h(α)) = h(α) h(α) −1 −1 −1 ❡ h(α)h(α ) = h(αα ) = h(s(α)) = s(h(α)) = h(α)h(α) . −1 2 −1

        ) ∈ G ) = ❈♦r♦❧ár✐♦ ✹✳✶✳✺✳ P❛r❛ q✉❛✐sq✉❡r (α, β ✱ t❡♠♦s q✉❡ h(αβ −1 1 h(α)h(β)

        ✳ ▲❡♠❛ ✹✳✶✳✻✳ P❛r❛ t♦❞♦ v ∈ E ✱ t❡♠♦s h(v) ∈ E ✳ 1 2

        ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ α ∈ G 2 t❛❧ q✉❡ h(v) = α✳ ❊♥tã♦ α = h(v) = h(vv) = h(v)h(v) = αα. ▲♦❣♦✱ α ∈ E ✳ 2

        ▲❡♠❛ ✹✳✶✳✼✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W 1 ✳ ❊♥tã♦✱ ♣❛r❛ q✉❛✐s✲ q✉❡r α, β ∈ W 1 ✱ t❡♠♦s q✉❡ r(α) = r(β) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r(h(α)) = r(h(β)) ✳

        −1 2

        ) ∈ G ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ r(α) = r(β)✳ ❊♥tã♦ (α, β ✳ −1 2 1

        )) ∈ G ❈♦♠♦ h é ❤♦♠♦♠♦r✜s♠♦✱ ❡♥tã♦ (h(α), h(β ✳ −1 −1 −1 2

        )) = r(h(β ) )) = r(h(β)) ▲♦❣♦✱ r(h(α)) = s(h(β ✱ ❝♦♠♦ q✉❡✲ rí❛♠♦s✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ r(α) 6= r(β)✳ ❊♥tã♦ h(r(α)) 6= h(r(β))✱ ♣♦✐s h é

        ❤♦♠♦♠♦r✜s♠♦ ✐♥❥❡t✐✈♦ ❡♠ W 1 ✳ ❊♥tã♦ r(h(α)) = h(r(α)) 6= h(r(β)) = r(h(β)).

        ) = W ▲❡♠❛ ✹✳✶✳✽✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W 1 ❡ q✉❡ h(W 1 2 ✳

        ❊♥tã♦✱ ♣❛r❛ t♦❞♦ α ∈ W 1 ✱ r(α) é ♣♦ç♦ s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r(h(α)) é ♣♦ç♦✳ ❉❡♠♦♥str❛çã♦✿ r(h(α)) 2 2 ∈ W 2 . ♥ã♦ é ♣♦ç♦ ⇐⇒ ❡①✐st❡ ❛r❡st❛ e t❛❧ q✉❡ h(α)e

        ∈ W ) = W ∈ W ) = e ❈♦♠♦ e 2 2 ❡ h(W 1 2 ✱ ❡♥tã♦ ❡①✐st❡ e 1 1 t❛❧ q✉❡ h(e 1 2 ✳

        P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱ ❡♥tã♦ h(α)e = h(α)h(e ) = h(αe ), 2 1 1 ♦ q✉❡ s✐❣♥✐✜❝❛ q✉❡ r(α) ♥ã♦ é ♣♦ç♦✳ 1 2

        ▲❡♠❛ ✹✳✶✳✾✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W ✳ ❊♥tã♦ (α, β) ∈ G 2 1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ (h(α), h(β)) ∈ G ✳ 2 2 ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ (α, β) /∈ G ✳ ❊♥tã♦ r(α) 6= s(β)✳ ❈♦♠♦ 1 h

        é ✐♥❥❡t✐✈❛ ❡♠ W 1 ✱ ❡♥tã♦ r(h(α)) = h(r(α)) 6= h(s(β)) = s(h(β)). 2 ▲♦❣♦✱ (h(α), h(β)) /∈ G ✳ 2 1 1 ) = W 2

        ▲❡♠❛ ✹✳✶✳✶✵✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W ❡ q✉❡ h(W ✳ 1 ❊♥tã♦✱ ♣❛r❛ q✉❛❧q✉❡r α ∈ W ✱ |α| = 1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ |h(α)| = 1✳ 1

        α ∈ W ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ e ∈ E ✳ ❙✉♣♦♥❤❛ q✉❡ h(e) = α 1 1 2 2 ✱ ❝♦♠

        α , α ∈ E / , f ∈ W ) = α ) = 1 2 ✳ ❊♥tã♦ ❡①✐st❡♠ f 2 1 2 1 t❛✐s q✉❡ h(f 1 1 ❡ h(f 2 α 2 ✳

        , f ∈ E / ◆♦t❡ q✉❡ f 1 2 ✱ ♣♦r ❊♥tã♦ 1 h(f f ) = h(f )h(f ) = α α = h(e). 1 2 1 2 1 2 f = e ∈ E ∈ E

        ▲♦❣♦✱ f 1 2 ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ f 1 ♦✉ f 1 2 ✱ ♦ q✉❡ é ✉♠ 1 ❛❜s✉r❞♦✳ ❆❣♦r❛✱ s✉♣♦♥❤❛ q✉❡ h(e) ∈ E ✳ ❊♥tã♦ 2 h(ee) = h(e)h(e) = h(e),

        ❧♦❣♦ e ∈ E ✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ 1 P♦r ♦✉tr♦ ❧❛❞♦✱ ❞❛❞♦ α ∈ W 1 s✉♣♦♥❤❛ q✉❡ |h(α)| = 1✳ ❙❡ α ∈ E ✱ 1 ❡♥tã♦ h(α) ∈ E ❡ |h(α)| = 0✳ ❙✉♣♦♥❤ ❡♥tã♦ q✉❡ |α| = n, n ≥ 2✳ ❊♥tã♦ 2 1

        α = α . . . α n i ∈ E 1 ✱ ♣❛r❛ α ✳ ❉❛í✱ 1 h(α) = h(α 1 . . . α n ) = h(α 1 ) . . . h(α n ), ) . . . h(α n )| = n

        ❡ ❡♥tã♦ 1 = |h(α)| = |h(α 1 ✱ ♦ q✉❡ é ✉♠ ❛❜s✉r❞♦✳ ) = W

        ❈♦r♦❧ár✐♦ ✹✳✶✳✶✶✳ ❙✉♣♦♥❤❛ h ✐♥❥❡t✐✈❛ ❡ q✉❡ h(W 1 2 ✳ ❊♥tã♦ |α| = |h(α)| .

        ✱ ♣❛r❛ t♦❞♦ α ∈ W 1 ) = W

        ▲❡♠❛ ✹✳✶✳✶✷✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W 1 ❡ q✉❡ h(W 1 2 ✳ → G

        ❊♥tã♦ h : G 1 2 é s♦❜r❡❥❡t♦r❛✳ . . . γ n

        ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ γ ∈ G i ∈ E ∪ (E ) ) = W ) = W 1 1 ∗ ∗ ∗ 2 ✳ ❊♥tã♦ ♣♦❞❡♠♦s ❡s❝r❡✈❡r γ = γ 1 ✱ ❡♠ q✉❡ γ ✳ ❈♦♠♦ h(W 2 2 1 2 ❡ h(W ✱ ❡♥tã♦ 1 1 2 1 ∗

        γ i = h(e i ) i ∈ W i ∈ E ∪ (E ) ✱ ♣❛r❛ ❝❡rt♦s e 1 ✳ P❡❧♦ ❧❡♠❛ e ✳ 1 1

        . . . e n P❡❧♦ ❧❡♠❛ e 1 é ❝♦♠♣♦♥í✈❡❧✱ ❧♦❣♦ h(e . . . e n ) = h(e ) . . . h(e n ) = γ . . . γ n = γ. 1 1 1

        ▲❡♠❛ ✹✳✶✳✶✸✳ h é ✐♥❥❡t✐✈❛ ❡♠ W ✳ 1 ∗ ∗ ∗ , β ∈

        ❉❡♠♦♥str❛çã♦✿ ❙❡❥❛♠ α, β ∈ W t❛✐s q✉❡ α 6= β✳ ❊♥tã♦ α ∗ ∗ ∗ ∗ 1 W 6= β ) 6= h(β ) 1 ❡ α ✳ ❈♦♠♦ h é ✶✲✶ ❡♠ W 1 ✱ s❡❣✉❡ q✉❡ h(α ✳ ❙❡♥❞♦ ❛ss✐♠✱ ∗ ∗ ∗ ∗ h(α) = h(α ) 6= h(β ) = h(β).

        ) = W ▲❡♠❛ ✹✳✶✳✶✹✳ ❙✉♣♦♥❤❛ q✉❡ h s❡❥❛ ✐♥❥❡t✐✈❛ ❡♠ W 1 ❡ q✉❡ h(W 1 2 ✳

        → G ❊♥tã♦ h : G 1 2 é ✐♥❥❡t✐✈❛✳ 1 ∗

        . . . e n ∈ G i ∈ E ∪ E ∪ (E ) ∪ ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦ g = e 1 ∗ 1 2 ✱ ❝♦♠ e 2 2 2 1

        (E ) i ∈ E ∪ E ∪ 2 ✳ P❡❧♦s ❧❡♠❛s ❡①✐st❡ ✉♠ ú♥✐❝♦ f 1 ∗ 1 1 (E ) ∪ (E ) i ) = e i 1 1 −1 t❛❧ q✉❡ h(f ✳

        (e . . . e n ) = f . . . f n . . . f n | = |e . . . e n | ❉❡✜♥❛ h 1 1 ✳ ◆♦t❡ q✉❡ |f 1 1 ✱

        . . . f n | &lt; |e . . . e n | . . . e n = h(f ) . . . h(f n ) ♣♦✐s s❡ |f 1 1 ✱ ❡♥tã♦ e 1 1 ♥ã♦ s❡r✐❛

        ❢♦r♠❛ r❡❞✉③✐❞❛✳

        ✹✳✷ ■s♦♠♦r✜s♠♦s ❣r❛❞✉❛❞♦s ❡♥tr❡ á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt

        ❉❡✜♥✐çã♦ ✹✳✷✳✶✳ ❙❡❥❛ G ✉♠ ❣r✉♣♦✐❞❡ ❡ s❡❥❛ R ✉♠ ❛♥❡❧ ❛ss♦❝✐❛t✐✈♦ ✭♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ✉♥✐t❛❧✮✳ ❉✐③❡♠♦s q✉❡ R é G✲❣r❛❞✉❛❞♦✱ ♦✉ ❣r❛❞✉❛❞♦ t ♣♦r G✱ s❡ R é ✉♠❛ s♦♠❛ ❞✐r❡t❛ ❞❡ s✉❜❣r✉♣♦s ❛❞✐t✐✈♦s R ✐♥❞❡①❛❞♦s ♣♦r t R s ⊆ R ts ❡❧❡♠❡♥t♦s ❞❡ G ❡ t❛✐s q✉❡ R ✱ ♣❛r❛ q✉❛✐sq✉❡r s, t ∈ G✳ M M

        A g B h ❉❡✜♥✐çã♦ ✹✳✷✳✷✳ ❙❡❥❛♠ A = ❡ B = á❧❣❡❜r❛s ❣r❛✲ g ∈G h ∈G 1 2

        ❞✉❛❞❛s ♣❡❧♦s ❣r✉♣♦✐❞❡s G 1 ❡ G 2 ✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❯♠ ❤♦♠♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦ ♣♦r ❣r✉♣♦✐❞❡ ❡♥tr❡ A ❡ B é ✉♠ ♣❛r (ϕ, h) ❡♠ q✉❡ ϕ : A → B

        → G é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛✱ h : G g ) ⊆ B h 1 2 é ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡

        ❣r✉♣♦✐❞❡ ❡ ϕ(A (g) ✱ ♣❛r❛ t♦❞♦ g ∈ G 1 ✳ → G

        Pr♦♣♦s✐çã♦ ✹✳✷✳✸✳ ❙❡ ❡①✐st❡ h : G 1 2 ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉✲ grad W ) = W ) ⋊ G = ∼ ♣♦✐❞❡ t❛❧ q✉❡ h| 1 é ✐♥❥❡t✐✈♦ ❡ h(W 1 2 ✱ ❡♥tã♦ D(X 1 1 D(X 2 ) ⋊ G 2 1 ) ⋊ G 1 → D(X 2 ) ⋊ G 2 e δ e ) = 1 h δ h v δ v ) = 1 h δ h ✱ ✈✐❛ ✉♠ ✐s♦♠♦r✜s♠♦ ϕ : D(X t❛❧ q✉❡ ϕ(1 (e) (e) ❡ ϕ(1 (v) (v) ✳ K (E )

        ❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ❡♠ L F 2 ❛ ❢❛♠í❧✐❛ 1 = {1 δ , 1 −1 δ −1 , 1 δ : v ∈ E , e ∈ E }. h (e) h (e) h h h (v) h (v) (e ) (e ) 1 1 ❱❛♠♦s ♠♦str❛r q✉❡ F s❛t✐s❢❛③ ❛s ❝♦♥❞✐çõ❡s ❞❡ á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❞❡

        L K (E 1 )

        ✭✐✮ 1 δ · 1 δ = 1 δ · 1 δ h (s(e)) h (s(e)) h (e) h (e) s (h(e)) s (h(e)) h (e) h (e) −1 = α (α (1 )1 )δ s s h s (h(e)) s (h(e)) (e) (h(e))h(e) (h(e)) = α s (1 s −1 (h(e)) (h(e)) (e) (s(e))h(e) 1 h )δ h = α s (1 s (h(e)) (h(e)) (e) (s(e)e) 1 h )δ h = α s (1 h )δ h (h(e)) (e) (e) = 1 h δ h . (e) (e) 1 h δ h · 1 r δ r = 1 h δ h · 1 h δ h (e) (e) (h(e)) (h(e)) (e) (e) (r(e)) (r(e)) −1

        = α (α (1 )1 )δ h (e) h (e) r (h(e)) h (e)r(h(e)) h (e) −1 = α h (1 h (e) (e) (h(e)) (er(e)) −1 1 r )δ h = α h (1 h )δ h (e) (e) (e) = 1 h δ h (e) (e)

        ✭✐✐✮

        1 h (r(e)) δ h (r(e)) · 1 h (e −1 ) δ h (e −1 ) = 1 r (h(e)) δ r (h(e)) · 1 h (e −1 ) δ h (e −1 ) = α r (h(e))−1 r (h(e)) (1 r (h(e)) )1 h (e) −1 )δ h (e −1 ) = α r (h(e)) (1 r (h(e))

        1 h (e −1 )h (e −1 ) = α r (h(e)) (1 h (e −1 )h (e −1 ) = 1 h (e −1 ) δ h (e −1 ) 1 h (e −1 ) δ h (e −1 ) · 1 s (h(e)) δ s (h(e)) = 1 h (e −1 ) δ h (e −1 ) · 1 s (h(e)) δ s (h(e))

        = α h (e −1 )−1 h (e −1 ) (1 h (e −1 ) )1 s (h(e))h (e −1 )s(h(e −1 )) = α h (e −1 ) (1 h (e −1 ) −1 1 s (h(e)) )δ h (e −1 s (e)) = α h (e −1 ) (1 h (e) 1 s (h(e)) )δ h (e −1 ) = α h (e −1 ) (1 h (e) )δ h (e −1 ) = 1 h (e −1 ) δ h (e −1 )

        ✭✐✐✐✮ 1 h (e −1 ) δ h (e −1 ) 1 h (g) δ h (g) = α h (e −1 )−1 h (e −1 ) (1 h (e −1 ) )1 h (g) )δ h (e −1 )h(g) = α h (e −1 ) (α e (1 h (e 1 )

        1 h (g) ))δ h (e −1 )h(g) = δ h (e),h(g)

        1 h (e) δ h (r(e)) = δ e,g

        1 h (e −1 ) δ h (r(e)) ✭✐✈✮ ❙❡❥❛ e ∈ E 1 t❛❧ q✉❡ 0 &lt; #{h(e) : s(h(e)) = h(v)} &lt; ∞✳ ◆♦t❡ q✉❡

        {e : s(e) = v} = {e : h(s(e)) = h(v)} = {e : s(h(e)) = h(v)} = {e : s(h(e)) = s(h(v))}. X s (e)=v = X s (e)=v

        1 h (e) δ h (e)

        1 h (e −1 ) δ h (e −1 ) = X s (e)=v

        1 h (e) δ s (h(e)) = X s (e)=v 1 h (e) δ s (h(e)) = X s (e)=v 1 h (e) δ h (v)

        = X e :s(h(e))=h(v)

        1 h (e) δ h (v) ❊♥tã♦✱ ❜❛st❛ ♠♦str❛r q✉❡ X e :s(h(e))=h(v)

        1 h (e) = 1 h (v) ✳

        ◆♦t❡ q✉❡ ♣❛r❛ ❝❛❞❛ e t❛❧ q✉❡ s(e) = h(v) ❡①✐st❡ ✉♠ ú♥✐❝♦ f t❛❧ 2 2 q✉❡ h(f) = e✳ ❆❧é♠ ❞✐ss♦✱ s❡ (h(v), e) ∈ G ❡♥tã♦ (v, f) ∈ G ✱ ♦✉ 2 1 s❡❥❛✱ s(f) = v✳ ◆♦t❡ t❛♠❜é♠ q✉❡ ♣❛r❛ ❝❛❞❛ e t❛❧ q✉❡ s(e) = v 2 t❡♠♦s q✉❡ (h(v), h(e)) ∈ G ✱ ❧♦❣♦✱ s(h(e)) = h(v)✳ P♦rt❛♥t♦✱ X 2 X e f :s(h(e))=h(v) :s(f )=h(v) 1 h = = 1 h . (e) (v) K (E 1 ) → L K (E 2 ) P♦rt❛♥t♦✱ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ϕ : L t❛❧

        δ ) = 1 δ −1 δ −1 ) = 1 −1 δ −1 δ ) = q✉❡ ϕ(1 e e h h ✱ ϕ(1 e e h h ❡ ϕ(1 v v (e) (e) (e ) (e ) 1 δ h (v) h (v) ✳ K (E ) H ❈♦♥s✐❞❡r❡ ❛❣♦r❛ ❡♠ L 1 ❛ ❢❛♠í❧✐❛ 1 = {1 −1 δ −1 , 1 −1 −1 δ −1 −1 , 1 −1 δ −1 : v ∈ E , e ∈ E }. h (e) h (e) h (e ) h (e ) h (v) h (v) −1 2 2 → G (W ) = W −1 ❈♦♠♦ h : G 1 2 é ✉♠❛ ❜✐❥❡çã♦ t❛❧ q✉❡ h 2 1 ❡ h | W = W 2 1 ✱ ♠♦str❛✲s❡ ❞❡ ♠♦❞♦ ❛♥á❧♦❣♦ q✉❡ ❡①✐st❡ ✉♠ ❤♦♠♦♠♦r✜s♠♦

        φ : L K (E ) → L K (E ) e δ e ) = 1 −1 δ −1 −1 δ −1 ) = 2 1 t❛❧ q✉❡ φ(1 h (e) h (e) ✱ ϕ(1 e e 1 −1 −1 δ −1 −1 v δ v ) = 1 −1 δ −1 h (e ) h (e ) ❡ ϕ(1 h (v) h (v) P♦r ✜♠✱ ♥♦t❡ q✉❡ φ é ❛ ✐♥✈❡rs❛ ❞❡ ϕ✳ ❱❛♠♦s ♠♦str❛r ❛❣♦r❛ ✉♠❛ r❡❝í♣r♦❝❛ ♣❛r❛ ❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r✳

        Pr♦♣♦s✐çã♦ ✹✳✷✳✹✳ ❙✉♣♦♥❤❛ q✉❡ E 1 s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦ ✭▲✮✳ ❙✉♣♦♥❤❛ ) ⋊ G → D(X ) ⋊ G q✉❡ ϕ : D(X 1 1 2 2 s❡❥❛ ✉♠ ✐s♦♠♦r✜s♠♦ ❡ q✉❡ h :

        G → G 1 2 s❡❥❛ ✉♠ ❤♦♠♦♠♦r✜s♠♦ ❞❡ ❣r✉♣♦✐❞❡ t❛✐s q✉❡ (ϕ, h) s❡❥❛ ✉♠ ✐s♦♠♦♠♦r✜s♠♦ ❣r❛❞✉❛❞♦✳ ❙✉♣♦♥❤❛ q✉❡ ϕ é t❛❧ q✉❡ W ϕ({1 v δ v : v ∈ E }) = {1 w δ w : w ∈ E }. 1 ) = S 1 2 2

        ❊♥tã♦ h| 1 é ✐♥❥❡t✐✈❛ ❡ h(S ✳ ) ⊆ E

        ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡✱ ♣❡❧♦ ❧❡♠❛ h(E 1 ) = S 2 1 2 ✳ ❆✜r♠❛çã♦ ✶✿ h(S ✳ 1 g δ g 6= 0 1 ) ⋊ G 1

        ❉❛❞♦ g ∈ S ✱ t❡♠♦s q✉❡ 1 ❡♠ D(X ✳ ▲♦❣♦ 0 6= ϕ(1 g δ g ) ∈ D h δ h (g) (g) ✱ ♣♦✐s ϕ é ✐♥❥❡t✐✈❛✱ ❡✱ ♣♦rt❛♥t♦✱ h(g) ∈ S ✳ 2 c δ c 6= 0 2 P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❥❛ c ∈ S ✳ ❊♥tã♦ 1 ✳ ❈♦♠♦ ϕ é ✐s♦♠♦r✜s♠♦✱

        δ δ = ϕ(x) = P ϕ(α δ ) ❡①✐st❡ x = P α g g t❛❧ q✉❡ 1 c c g g ✳ g δ g ) ∈ D h δ h c δ c = ϕ(α g δ g ) 6= X

        ❈♦♠♦ ϕ(α (g) (g) ✱ ♣❛r❛ t♦❞♦ g✱ ❡♥tã♦ 1 g δ g ) 6= 0 g δ g 6= 0 h (g)=c ✳ ▲♦❣♦✱ ❡①✐st❡ g t❛❧ q✉❡ h(g) = c ❡ ϕ(α ✱ ♣♦rt❛♥t♦ α ❡ g ∈ S 1 ✳ 1

        ❆✜r♠❛çã♦ ✷✿ h| E é ✐♥❥❡t✐✈❛✳

        ❙❡❥❛♠ v, w ∈ E 1 ✳ P♦r ❤✐♣ót❡s❡✱ ϕ(1 v δ v 1 w δ w ) = ϕ(1 v δ v )ϕ(1 w δ w ) = 1 h (v) δ h (v) ·1 h (w) δ h (w) = δ h (v),h(w) 1 h (v) δ h (v) .

        ✉♠ s✉❜❝❛♠✐♥❤♦ α ❞❡ α ✭♣♦ss✐✈❡❧♠❡♥t❡ ♥✉❧♦✱ ❝❛s♦ s(e) = s(α)✮ t❛❧ q✉❡ α e

        ❡ r(β) = r(γ)✱ ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✳ ❉❛í✱ 1 β = 1 α = 1 α −1 = 1 γ ,

        ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡ α = βγ −1 ✱ ❡♠ q✉❡ β, γ ∈ W 1 \ E 1

        = 1 α −1 ✱ ❡ t❡♠♦s ✉♠ ❛❜s✉r❞♦✳

        e) = 1 ✱ ♣♦ré♠✱ 1 s (α)

        (α

        e) = 0 ❡ 1 s (α)

        é ❝♦♠♣♦♥í✈❡❧ ❡ ♥ã♦ ❝♦♥té♠ α ❝♦♠♦ s✉❜❝❛♠✐♥❤♦✳ ❊♥tã♦ 1 α (α

        (L) ❡ α é ✉♠ ❝✐❝❧♦✱ ❡①✐st❡ ✉♠❛ s❛í❞❛ e ∈ E 1 1 ❞❡ α ❡ ❡♥tã♦ ♣♦❞❡♠♦s t♦♠❛r

        P♦r ♦✉tr♦ ❧❛❞♦✱ ϕ(1 v δ v 1 w δ w ) = ϕ(δ v,w 1 v δ v ) = δ v,w 1 h (v) δ h (v) . P♦rt❛♥t♦✱ h(v) = h(w) s❡✱ ❡ s♦♠❡♥t❡ s❡✱ v = w✳ ❆✜r♠❛çã♦ ✸✿ ❙❡❥❛ α ∈ S 1 ✳ ❙❡ h(α) ∈ E 2 ❡♥tã♦ α ∈ E 1 ✳

        P♦rt❛♥t♦✱ r(α) = s(α) ❡ α = α −1 ✳ Pr✐♠❡✐r♦ ✈❛♠♦s s✉♣♦r q✉❡ α ∈ W 1 ♦✉ α ∈ W 1 ✳ ❚r♦❝❛♥❞♦ α ♣♦r α −1 s❡ ♥❡❝❡ssár✐♦✱ ✈❛♠♦s ❛ss✉♠✐r q✉❡ α ∈ W 1 ✳ ❈♦♠♦ E 1 s❛t✐s❢❛③ ❛ ❝♦♥❞✐çã♦

        ✱ ♦✉ s❡❥❛✱ 1 α δ s (α) = 1 α −1 δ r (α) .

        ❆♥❛❧♦❣❛♠❡♥t❡✱ ϕ(1 α −1 δ α −1 1 α δ α ) = baδ h (α) = abδ h (α) ✳ ❈♦♠♦ ϕ é ✐s♦♠♦r✜s♠♦✱ s❡❣✉❡ q✉❡ 1 α δ α 1 α −1 δ α −1 = 1 α −1 δ α −1 1 α δ α

        = D h −1 ) sã♦ t❛✐s q✉❡ ϕ(1 α δ α ) = aδ h (α) ❡ ϕ(1 α −1 δ α −1 ) = bδ h −1 ) = bδ h (α) ✳

        1 α −1 δ α −1 ) = ϕ(1 α δ α )ϕ(1 α −1 δ α −1 ) = aδ h (α)h (α −1 ) = abδ h (α) , ❡♠ q✉❡ a, b ∈ D h (α)

        ϕ(1 α δ α

        P❡❧❛ ❛✜r♠❛çã♦ ❛♥t❡r✐♦r✱ s(α) = r(α)✳ ❱❛♠♦s ♠♦str❛r q✉❡ α ∈ E 1 ✳ ❙✉♣♦♥❤❛✱ ♣♦r ❛❜s✉r❞♦✱ ♦ ❝♦♥trár✐♦✳ ❉❛í✱

        ❱❛♠♦s ♠♦str❛r q✉❡ s❡ α /∈ E 1 ❡♥tã♦ h(α) /∈ E 2 ✳ ◆♦t❡ q✉❡✱ ♣❡❧♦ ❧❡♠❛ ❡ ❝♦♠♦ h(α) ∈ E 2 ✱ ❡♥tã♦ h(s(α)) = s(h(α)) = r(h(α)) = h(r(α)).

        ❝♦♥tr❛❞✐③❡♥❞♦ ♦ ❢❛t♦ ❞❡ q✉❡ α = βγ −1 ❡stá ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✳ P♦rt❛♥t♦✱ α ∈ E 1 ✳ ❆✜r♠❛çã♦ ✹✿ h é ✐♥❥❡t✐✈♦ ❡♠ W 1 ✳ ❙❡❥❛♠ α, β ∈ W 1 t❛✐s q✉❡ h(α) = h(β)✳ ❊♥tã♦ h(r(α)) = r(h(α)) = r(h(β)) = h(r(β)).

        −1

        P❡❧❛ ❛✜r♠❛çã♦ ✷✱ t❡♠♦s q✉❡ r(α) = r(β)✳ ❙❡♥❞♦ ❛ss✐♠✱ αβ é ❝♦♠✲ ♣♦♥í✈❡❧ ❡ −1 −1 −1 h(αβ ) = h(α)h(β) = h(α)h(α) = s(α) ∈ E . −1 2

        ∈ E P❡❧❛ ❛✜r♠❛çã♦ ✸✱ αβ ✳ P♦rt❛♥t♦✱ α = β✳ 1 Pr♦♣♦s✐çã♦ ✹✳✷✳✺✳ ❙♦❜ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r ❡ e δ e : e ∈ E }) = {1 f δ f : f ∈ E } ) = W 1 1 s✉♣♦♥❞♦ q✉❡ ϕ({1 ✱ ❡♥tã♦ h(W 1 2 1 2 ✳

        δ ) = 1 δ ❉❡♠♦♥str❛çã♦✿ ◆♦t❡ q✉❡ ❝♦♠♦ ϕ é ❣r❛❞✉❛❞♦✱ ❡♥tã♦ ϕ(1 e e h h ✱ 1 (e) (e)

        ♣❛r❛ t♦❞♦ e ∈ E ✳ 1 1 e δ e : e ∈ E }) = {1 f δ f : f ∈ E } 1 1 ❙❡❥❛ f ∈ E ✳ ❈♦♠♦ ϕ({1 ✱ ❡♥tã♦ 1 2 f δ f = ϕ(1 e δ e ) e δ e ) = 1 h δ h 1 2

        ❡①✐st❡ e ∈ E t❛❧ q✉❡ 1 ✳ ❈♦♠♦ ϕ(1 (e) (e) ✱ ❡♥tã♦ 1 h(e) = f . . . f n ∈ W , . . . , e n ∈ W ✳ ❉❛❞♦ f 1 2 ✱ ❡♥tã♦ ❡①✐st❡♠ e 1 1 t❛✐s q✉❡ h(e i ) = f i

        ✳ ▲♦❣♦✱ ♣❡❧♦ ❧❡♠❛ f 1 . . . f n = h(e 1 ) . . . h(e n ) = h(e 1 . . . e n ). ⊆ h(W )

        P♦rt❛♥t♦✱ W 2 1 ✳ . . . e n ∈ W . . . e n ) = h(e ) . . . h(e n )

        ❙❡❥❛ e 1 1 1 ✳ ❊♥tã♦ h(e 1 1 ❀ ❝♦♠♦ h(e i ) ∈ E . . . e n ) ∈ W 2 ✱ ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✱ ❡♥tã♦ h(e 1 2 ✳ ▲♦❣♦✱ h(W 1 ) ⊆ W 2

        ❱❡❥❛♠♦s ❛❣♦r❛ q✉❡ ❛ ❝♦♥❞✐çã♦ ❛❞✐❝✐♦♥❛❧ ♥❛ ♣r♦♣♦s✐çã♦ ❛♥t❡r✐♦r é ) = W

        ♥❡❝❡ssár✐❛ ♣❛r❛ ♦❜t❡r♠♦s h(W 1 = {E , E , r 1 1 2 ✳ 1 , s 1 } = {v 1 , v 2 , v 3 } ❊①❡♠♣❧♦ ✹✳✷✳✻✳ ❙❡❥❛ E 1 1 1 ✱ ❡♠ q✉❡ E 1 ✱

        E = {e 1 ✱ r ✱ s ❡ r ✳ P♦❞❡✲ 1 , e 2 } 1 (e 1 ) = v 2 = s 1 (e 2 ) 1 (e 1 ) = v 1 1 (e 2 ) = v 3 ♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❣r❛❢♦ ♣♦r e e 1 2 1 v // v // v 1 2 3 1

        = {E , E , r , s } = {w , w , w } = {f , f } ❙❡❥❛ E 2 2 2 2 2 ✱ ❡♠ q✉❡ E 2 1 2 3 ✱ E 2 1 2 ✱ r (f ) = w = r (f ) (f ) = w (f ) = w 2 1 3 2 2 ✱ s 2 1 1 ❡ s 2 2 2 ✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r

        ❡st❡ ❣r❛❢♦ ♣♦r w 3 f 1 == aa f 2 w w 1 −1 −1 −1 −1 2 = {v , v , v , e , e , e , e , e e , e e }

        ◆♦t❡ q✉❡ G 1 1 2 3 1 2 1 2 1 2 ✳ 2 1

        ❉❡✜♥❛ h : G 1 → G 2 ❞❡ ♠♦❞♦ q✉❡ h(e 1 ) = f −1 1

        ✱ h(e 2 ) = f 1 f −1 2

        ✱ h(e −1 1 ) = h(e 1 ) −1 ❡ h(e −1 2

        ) = h(e 2 ) −1 ✳ ◆♦t❡ q✉❡ h(v 1 ) = h(e 1 e −1 1 ) = h(e 1 )h(e 1 ) −1 = f −1 1 f 1 = r(f 1 ) = w 3 h(v 2 ) = h(e −1 1 e 1 ) = h(e 1 ) −1 h(e 1 ) = f 1 f −1 1 = s(f 1 ) = w 1 h(v 3 ) = h(e −1 2 e 2 ) = h(e 2 ) −1 h(e 2 ) = (f 1 f −1 2 ) −1 (f 1 f −1 2 )

        = f 2 r(f 1 )f −1 2 = s(f 2 ) = w 2 .

        ➱ ❢á❝✐❧ ✈❡r q✉❡ ❡①✐st❡ ✉♠ ✐s♦♠♦r✜s♠♦ ϕ : L K (E 1 ) → L K (E 2 ) t❛❧ q✉❡

        ϕ(1 e 1 δ e 1 ) = 1 f −1 1 δ f −1 1 ✱ ϕ(1 e 2

        δ e 2 ) = 1 f 1 f −1 2 δ f 1 f −1 2 ✱ ϕ(1 v 1

        δ v 1 ) = 1 w 3 δ w 3 ✱ ϕ(1 v 2 δ v 2 ) = 1 w 1 δ w 1 ✱ ϕ(1 v 3 δ v 3 ) = 1 w 2 δ w 2 ❡ q✉❡ (ϕ, h) é ✉♠ ✐s♦♠♦r✜s♠♦

        ❣r❛❞✉❛❞♦✳ ◆♦t❡ q✉❡ ϕ({1 v 1

        δ v 1 , 1 v 2 δ v 2 , 1 v 3 δ v 3 }) = {1 w 1 δ w 1 , 1 w 2 δ w 2 , 1 w 3 δ w 3 } ✱ ♠❛s

        ϕ({1 e 1 δ e 1 , 1 e 2 δ e 2 }) 6= {1 f 1 δ f 1 , 1 f 2 δ f 2 } ✳ ❖✉ s❡❥❛✱ ❛ ❝♦♥❞✐çã♦ ❛❞✐❝✐♦♥❛❧ ❞❛

        ♣r♦♣♦s✐çã♦ ♥ã♦ é s❛t✐s❢❡✐t❛ ❡ t❡♠♦s✱ ❝❧❛r❛♠❡♥t❡✱ h(W 1 ) 6= W 2

        ❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s

        ❬✶❪ ▼✳ ❉♦❦✉❝❤❛❡✈ ❡ ❘✳ ❊①❡❧✱ ❆ss♦❝✐❛t✐✈✐t② ♦❢ ❝r♦ss❡❞ ♣r♦❞✉❝ts ❜② ♣❛r✲ t✐❛❧ ❛❝t✐♦♥s✱ ❡♥✈❡❧♦♣✐♥❣ ❛❝t✐♦♥s ❛♥❞ ♣❛rt✐❛❧ r❡♣r❡s❡♥t❛t✐♦♥s✱ ❚❛♥s✳ ❆♠❡r✳ ▼❛t❤✳ ❙♦❝✳✱ ✸✺✼✱ ✭✷✵✵✺✮✱ ✶✾✸✶✲✶✾✺✷

        ❬✷❪ ❉✳ ●♦♥ç❛❧✈❡s ❡ ❉✳ ❘♦②❡r✱ ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s ❛s ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣ r✐♥❣s✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ✐♥ ❆❧❣❡❜r❛✱ ✈✳ ✹✷✱ ♣✳ ✸✺✼✽✲✸✺✾✷✱ ✷✵✶✹✳

        ❬✸❪ ❉✳ ●♦♥ç❛❧✈❡s✱ ❏✳ Ö✐♥❡rt ❡ ❉✳ ❘♦②❡r✱ ❙✐♠♣❧✐❝✐t② ♦❢ ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣ r✐♥❣s ✇✐t❤ ❛♣♣❧✐❝❛t✐♦♥s t♦ ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s ❛♥❞ t♦♣♦❧♦❣✐❝❛❧ ❞②✲ ♥❛♠✐❝s✱ ❏♦✉r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ✭Pr✐♥t✮✱ ✈✳ ✹✷✵✱ ♣✳ ✷✵✶✲✷✶✻✱ ✷✵✶✹✳

        ❬✹❪ ❉✳ ●♦♥ç❛❧✈❡s✱ ❙✐♠♣❧✐❝✐t② ♦❢ ♣❛rt✐❛❧ s❦❡✇ ❣r♦✉♣ r✐♥❣s ♦❢ ❛❜❡❧✐❛♥ ❣r♦✉♣s✱ ❈❛♥❛❞✐❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❇✉❧❧❡t✐♥✱ ✈✳ ✺✼✱ ♣✳ ✶✲✶✶✱ ✷✵✶✹✳

        ❬✺❪ ●✳ ❇♦❛✈❛✱ ❈❛r❛❝t❡r✐③❛çõ❡s ❞❛ ❈✯✲á❧❣❡❜r❛ ❣❡r❛❞❛ ♣♦r ✉♠❛ ❝♦♠♣r❡s✲ sã♦ ❛♣❧✐❝❛❞❛s ❛ ❝r✐st❛✐s ❡ q✉❛s✐❝r✐st❛✐s ✲ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮✱ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ✭✷✵✵✼✮✳

        ❬✻❪ ❉✳ ●♦♥ç❛❧✈❡s✱ Pr♦❞✉t♦s ❈r✉③❛❞♦s ✲ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮✱ ❯♥✐✲ ✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❙❛♥t❛ ❈❛t❛r✐♥❛✱ ✭✷✵✵✶✮✳

        ❬✼❪ ▼✳ ❚♦♠❢♦r❞❡✱ ❯♥✐q✉❡♥❡ss t❤❡♦r❡♠s ❛♥❞ ✐❞❡❛❧ str✉❝t✉r❡ ❢♦r ▲❡❛✈✐tt ♣❛t❤ ❛❧❣❡❜r❛s✱ ✭✷✵✵✼✮✱ ❏✳ ❆❧❣❡❜r❛✱ ✸✶✽✱ ✷✼✵✲✷✾✾

        ❬✽❪ ❉✳ ❇❛❣✐♦✱ ❉✳ ❋❧♦r❡s✱ ❆✳ P❛q✉❡s✱ P❛rt✐❛❧ ❛❝t✐♦♥s ♦❢ ♦r❞❡r❡❞ ❣r✉♣♦✐❞s ♦♥ r✐♥❣s✱ ✭✷✵✶✵✮✱ ❏♦r♥❛❧ ♦❢ ❆❧❣❡❜r❛ ❛♥❞ ✐ts ❛♣♣❧✐❝❛t✐♦♥s✱ ✈♦❧ ✾ ✐ss✉❡ ✸✳

        ❬✾❪ ❉✳ ❇❛❣✐♦✱ ❆✳ P❛q✉❡s P❛rt✐❛❧ ❣r✉♣♦✐❞ ❛❝t✐♦♥s✿ ❣❧♦❜❛❧✐③❛t✐♦♥✱ ▼♦r✐t❛ t❤❡♦r② ❛♥❞ ●❛❧♦✐s t❤❡♦r②✱ ❈♦♠♠✉♥✐❝❛t✐♦♥s ■♥ ❆❧❣❡❜r❛ ✭❖♥❧✐♥❡✮✱ ✈✳ ✹✵✱ ♣✳ ✸✻✺✽✲✸✻✼✽✱ ✷✵✶✷✳

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