Produtos cruzados parciais algébricos e aplicação à Álgebra de Leavitt

❯♥✐✈❡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

❯♥✐✈❡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❛✳

Ficha de identificação da obra elaborada pelo autor,

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

Yoneda, Gabriela Silmaia da Silva

Produtos cruzados parciais algébricos e aplicação à Álgebra de Leavitt / Gabriela Silmaia da Silva Yoneda ; orientador, Daniel Gonçalves - Florianópolis, SC, 2015. 69 p.

Dissertação (mestrado) - Universidade Federal de Santa Catarina, Centro de Ciências Físicas e Matemáticas. Programa de Pós-Graduação em Matemática Pura e Aplicada.

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❡✳

❘❡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❛✲ ❞♦s ♣❛r❝✐❛✐s ❡ s❡✉s r❡s♣❡❝t✐✈♦s ✐s♦♠♦r✜s♠♦s ❝♦♠ LK(E)✳ ❆❧é♠ ❞✐ss♦✱

❡st✉❞❛♠♦s ❝♦♥❞✐çõ❡s s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡ ❞❛❞♦s ❞♦✐s ❣r❛❢♦s ❞✐r✐❣✐❞♦sE1

❡E2✱ ❝♦♥s✐❞❡r❛♥❞♦ s❡✉s ❣r✉♣♦✐❞❡sG1❡G2✱ 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✳

❆❜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❤

❛❧❣❡❜r❛ LK(E)❛♥❞ 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❛♣❤sE1❛♥❞E2✱ ❝♦♥s✐❞❡r✐♥❣ t❤❡✐r ❣r♦✉♣♦✐❞sG1❛♥❞G2✱

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✳

❙✉♠ár✐♦

✶ Pr♦❞✉t♦s ❝r✉③❛❞♦s ♣❛r❝✐❛✐s ❡ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡ ❛çõ❡s ♣❛r✲

❝✐❛✐s ❞❡ ❣r✉♣♦ ✺

✶✳✶ ❆çõ❡s ♣❛r❝✐❛✐s ❞❡ ✉♠ ❣r✉♣♦G✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺

✶✳✷ Pr♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾ ✷ ❆s á❧❣❡❜r❛s ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt ❝♦♠♦ ♣r♦❞✉t♦ ❝r✉✲

③❛❞♦ ♣❛r❝✐❛❧ ✶✹

✷✳✶ ❆çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ✶✹ ✷✳✷ ➪❧❣❡❜r❛s ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✺ ✸ LK(E) ❝♦♠♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦✐❞❡ ✸✵

✸✳✶ ●r✉♣♦✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵ ✸✳✷ ❆çã♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✸✳✸ ➪❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❝♦♠♦ ♣r♦❞✉t♦ ♣❛r❝✐❛❧ ❞❡ ❣r✉♣♦✐❞❡ ✳ ✳ ✸✽ ✹ ■s♦♠♦r✜s♠♦s ❡♥tr❡ á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt ✹✼ ✹✳✶ ❍♦♠♦♠♦r✜s♠♦s ❞❡ ❣r✉♣♦✐❞❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✼ ✹✳✷ ■s♦♠♦r✜s♠♦s ❣r❛❞✉❛❞♦s ❡♥tr❡ á❧❣❡❜r❛s ❞❡ ▲❡❛✈✐tt ✳ ✳ ✳ ✳ ✺✶

■♥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❛ ❞❡ ▲❡✲ ❛✈✐tt LK(E) ❞❡ ✉♠ ❣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 ❞❡ ❝❛♠✐♥❤♦s ❞❡ ▲❡❛✈✐tt✱ LK(E)✱ ❡ à ❡①✐stê♥❝✐❛ ❞❡ ✉♠ ✐s♦♠♦r✜s♠♦ ❡♥tr❡ LK(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 G1 ❡G2 ❛ss♦❝✐❛❞♦s ❛ ❣r❛❢♦s E1 ❡E2✱ 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✉♣♦Gs♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦ X é ✉♠ ♣❛r α= ({Dt}t∈G,{αt}t∈G) ❡♠ q✉❡✱ ♣❛r❛ ❝❛❞❛ t ∈ G✱ Dt é

✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡G❡αt:Dt−1 →D_ts❛t✐s❢❛③✿ ✭✐✮ De=X✱ ❡♠ q✉❡e∈Gé ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡G❀

✭✐✐✮ α−_t1(Dt∩Ds−1)⊆D_(st)−1❀

✭✐✐✐✮ αs◦αt(x) =αst(x)✱ ♣❛r❛ t♦❞♦x∈α−t1(Dt∩Ds−1)

❖❜s❡r✈❛çã♦ ✶✳✶✳✷✳ ◆♦t❡ q✉❡ ❛ ✐❣✉❛❧❞❛❞❡ ❡♠(iii)❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣♦✐s✱

s❡x∈α−t1(Dt∩Ds−1)✱ ❡♥tã♦ α_t(x) ❢❛③ s❡♥t✐❞♦ ❡α_t(x)∈ D_t∩D_s−1✳

❙❡♥❞♦ ❛ss✐♠✱ ♣♦❞❡♠♦s ❛♣❧✐❝❛r αs ❡♠ αt(x)✳ ❆❧é♠ ❞✐ss♦✱ ♦ ✐t❡♠ (ii)

❣❛r❛♥t❡ q✉❡ ♣♦❞❡♠♦s ❛♣❧✐❝❛rαst❡♠xs❡♠♣r❡ q✉❡x∈αt−1(Dt∩Ds−1)✳ ❖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

é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ α= ({Dt}t∈G,{αt}t∈G) ❞♦ ❣r✉♣♦ Gt❛❧ q✉❡ Dt é

✉♠ ✐❞❡❛❧ ❞❡R ❡αt:Dt−1 →D_t é ✉♠ ✐s♦♠♦r✜s♠♦ ❞❡ ❛♥❡❧✱ ♣❛r❛ t♦❞♦

t∈G✳

Pr♦♣♦s✐çã♦ ✶✳✶✳✹✳ ❆s ❝♦♥❞✐çõ❡s ❞❡ ✶✳✶✳✶ sã♦ ❡q✉✐✈❛❧❡♥t❡s ❛✿ ✭✐✮ De=X ❡αe= IdX❀

✭✐✐✮ αt(Dt−1∩D_s) =D_t∩D_ts❀

✭✐✐✐✮ αs(αt(x)) =αst(x)✱ ♣❛r❛ t♦❞♦x∈Dt−1∩D_(st)−1

❊①❡♠♣❧♦ ✶✳✶✳✺✳ ❙❡❥❛Z♦ ❣r✉♣♦ ❛❞✐t✐✈♦ ❞♦s ✐♥t❡✐r♦s ❡ s❡❥❛N♦ ❝♦♥❥✉♥t♦

❞♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❉❡✜♥❛✱ ♣❛r❛ ❝❛❞❛z∈Z✱

Dz={n∈N:n≥z}.

◆♦t❡ q✉❡Dz=N✱ ❝❛s♦z≤0✳ ❉❡✜♥❛✱ ♣❛r❛ ❝❛❞❛z∈Z✱ αz: D−z → Dz

n 7→ n+z✳

❱❡❥❛ q✉❡ αz ❡stá ❜❡♠ ❞❡✜♥✐❞❛ ♣❛r❛ t♦❞♦ z ∈ Z ♣♦✐s✱ s❡ z ≤ 0

❡ n ∈ D−z✱ ❡♥tã♦ n ≥ −z ❡ αz(n) = n+z ≥ −z+z = 0✳ ▲♦❣♦✱ αz(n)∈N=Dz✳ ❙❡z >0 ❡n∈D−z=N✱ ❡♥tã♦αz(n) =n+z≥z❡

❝♦♠ ✐ss♦αz(n)∈Dz✳

◆♦t❡ q✉❡✱ ♣❛r❛ t♦❞♦ z ∈Z ❡ ♣❛r❛ t♦❞♦_{n∈ Dz}✱ _{αz◦α−z(n) =z}✳

▲♦❣♦✱α−1

z =α−z✳

❱❛♠♦s ♠♦str❛r q✉❡α= ({Dz}z∈Z,{αz}z∈Z)é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞♦ ❣r✉♣♦Zs♦❜r❡ ♦ ❝♦♥❥✉♥t♦N✳

✭✐✮ P♦r ❞❡✜♥✐çã♦✱D0=N✳

✭✐✐✮ P❛r❛ ♠♦str❛r q✉❡α−1

z (Dz∩D−w)⊆D−z−w ❞❡✈❡♠♦s ❝♦♥s✐❞❡r❛r

❞♦✐s ❝❛s♦s✳

❈❛s♦ ✶✿ −z−w ≤0✳ ◆❡st❡ ❝❛s♦✱ D−z−w =N❡ ❡♥tã♦ s❡❣✉❡

❞✐r❡t❛♠❡♥t❡ q✉❡ Dz∩D−w⊆D−z−w✳

❈❛s♦ ✷✿ −z−w >0✳ ◆❡st❡ ❝❛s♦✱ t❡♠♦s z <−w✱ ♦ q✉❡ ✐♠♣❧✐❝❛

❡♠Dz∩D−w=D−w✳ ❉❛í✱ ♣❛r❛x∈D−w✱ t❡♠♦s x≥ −w❡α−z(x) =x−z≥ −w−z.

▲♦❣♦✱α−z(x)∈D−w−z✳

P♦rt❛♥t♦✱α−z(Dz∩D−w)⊆D−z−w✳

✭✐✐✐✮ ❙❡❥❛♠z, w∈Z❡ s❡❥❛_n∈α−1

z (Dz∩D−w)✳ ❊♥tã♦

αw◦αz(n) =αw(z+n) =w+ (z+n) = (w+z) +n=αw+z(n).

❊①❡♠♣❧♦ ✶✳✶✳✻✳ ❙❡❥❛ α= ({Dg}g∈G,{αg}g∈G) ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡

✉♠ ❣r✉♣♦Gs♦❜r❡ ✉♠ ❝♦♥❥✉♥t♦X ❡ s❡❥❛ K✉♠ ❝♦r♣♦ q✉❛❧q✉❡r✳

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

P❛r❛ t♦❞❛f ∈Eg−1✱ ♣❛r❛ t♦❞♦x∈X✱ s❡❥❛

βg(f)(x) =

(

f(αg−1(x)) ✱ s❡x∈D_g

0 ✱ s❡x /∈Dg.

❱❡❥❛ q✉❡ βg é ✉♠❛ ❢✉♥çã♦βg:Eg−1 →E_g✳

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

✭✐✮ ◆♦t❡ q✉❡

Ee={f ∈KX :f|X\De = 0}

={f ∈KX _:f| X\X=0}

=KX_.

✭✐✐✮ P❛r❛ q✉❛✐sq✉❡r g, h∈G ❡f ∈(Eg−1∩E_h)✱ ✈❛♠♦s ♠♦str❛r q✉❡

βg(f)∈Egh✳ ❙❡❥❛x∈X\Dgh✳ ❚❡♠♦s ❞♦✐s ❝❛s♦s✳

❈❛s♦ ✶✿ x /∈Dg✳

❊♥tã♦βg(f)(x) = 0✱ ♣♦r ❞❡✜♥✐çã♦✳

❈❛s♦ ✷✿ x∈Dg✳

◆❡st❡ ❝❛s♦✱ t❡♠♦s q✉❡x∈Dg\Dgh=Dg\(Dgh∩Dg)✳ ▲♦❣♦✿ αg−1(x)∈α_g−1(D_g\(D_g∩D_gh)) =D_g−1\α_g−1(D_g∩D_gh)

=Dg−1\(D_g−1∩D_h).

❖✉ s❡❥❛✱αg−1(x)∈D_g−1\(D_g−1∩D_h)⊆X\D_h✳ ▲♦❣♦✱ βg(f)(x) = f(αg−1(x)) = 0✱ ♣♦✐s f|_X\D

h = 0✱ ✈✐st♦ q✉❡ f ∈Eh✳ P♦rt❛♥t♦✱βg(f)|X\Dgh = 0✳ ■st♦ é✱βg(Eg−1∩Eh)⊆Egh✳

✭✐✐✐✮ P❛r❛ q✉❛✐sq✉❡rg, h∈G, f ∈Eh−1∩E_h−1_g−1 ❡x∈X✱ t❡♠♦s

βg(βh(f))(x) =

(

βh(f)(αg−1(x)) ✱ s❡x∈D_g

0 ✱ s❡x /∈Dg

=     

f(αh−1(α_g−1(x))) ✱ s❡x∈D_g , α_g−1(x)∈D_h

0 ✱ s❡x∈Dg , αg−1(x)∈/D_h

0 ✱ s❡x /∈Dg.

◆♦t❡ q✉❡

x∈Dg ❡αg−1(x)∈D_h⇔x∈D_g ❡α_g−1(x)∈D_h∩D_g−1

⇔x∈Dg ❡x=αg(αg−1(x))∈α_g(D_h∩D_g−1)

⇔x∈Dg ❡x∈D(g−1₎−1∩D_(g−1₎−1_(h−1₎−1

⇔x∈Dg∩Dgh.

▲♦❣♦✱

βg(βh(f))(x) =

(

f(αh−1(α_g−1(x))) ✱ s❡x∈D_g∩D_gh

0 ✱ ❝❛s♦ ❝♦♥trár✐♦

=     

f(αh−1_g−1(x)) ✱ s❡x∈D_gh∩D_g

0 ✱ s❡x∈Dgh\Dg 0 ✱ s❡x /∈Dgh.

❱❡❥❛ q✉❡ s❡x∈Dgh\Dg=Dgh\(Dgh∩Dg)✱ s❡❣✉❡ q✉❡

αh−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)

=Dh−1_g−1\(D_h−1_g−1∩D_h−1)

=Dh−1_g−1\D_h−1 ⊆X\D_h−1

❈♦♠♦f ∈Eh−1✱ ♦❜t❡♠♦sf(α_h−1_g−1) = 0✳ ❉✐ss♦ s❡❣✉❡ q✉❡

βg(βh(f))(x) =

  

 

f(αh−1_g−1(x)), s❡x∈D_g∩D_gh

f(αh−1_g−1(x)), s❡x∈D_gh\D_g

0, s❡x /∈Dgh

=

(

f(αh−1_g−1(x)), s❡x∈D_gh

0 ✱ s❡x /∈Dgh.

=βgh(f)(x).

❈♦♠♦ ♦✉tr♦s ❝❛s♦s sã♦ ❡q✉✐✈❛❧❡♥t❡s✱ t❡♠♦s q✉❡

βg(βh(f))(x) =

(

f(α(gh)−1(x)) ✱ s❡x∈D_gh

0 ✱ s❡x /∈Dgh.

P♦rt❛♥t♦✱ ♣❛r❛ q✉❛✐sq✉❡r g, h∈G❡f ∈Dh−1∩D_h−1_g−1✱ t❡♠♦s

βg(βh(f)) =βgh(f).

✶✳✷ Pr♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧

❙❡❥❛A✉♠❛ á❧❣❡❜r❛ s♦❜r❡ ✉♠ ❝♦r♣♦K❡ s❡❥❛α= ({Dt}t∈G,{αt}t∈G)

✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦Gs♦❜r❡ ❛ á❧❣❡❜r❛ A✳ ❈♦♥s✐❞❡r❡ ♦ ❝♦♥✲

❥✉♥t♦ ❞❡ t♦❞❛s ❛s s♦♠❛s ❢♦r♠❛✐s ✜♥✐t❛s ❞❛ ❢♦r♠❛X

g∈G

agδg✱ ❝♦♠ag∈Dg✳

❈♦♥s✐❞❡r❡ ❛s s❡❣✉✐♥t❡s ♦♣❡r❛çõ❡s ♥♦ ❝♦♥❥✉♥t♦ ❞❡✜♥✐❞♦ ❛❝✐♠❛✿

❼ X

g∈G agδg+

X

g∈G bgδg =

X

g∈G

(ag+bg)δg;

❼ λX g∈G

agδg =

X

g∈G

(λag)δg✱ ❡♠ q✉❡λ∈K❀

❼ (agδg)·(bhδh) = αg(α−g1(ag)bh)δgh. ❊ ❡ss❛ ♦♣❡r❛çã♦ ❡st❡♥❞❡✲s❡

❧✐♥❡❛r♠❡♥t❡ ♣❛r❛ t♦❞♦ ♦ ❝♦♥❥✉♥t♦✳ ❉❡✜♥✐çã♦ ✶✳✷✳✶✳ ❖ ❝♦♥❥✉♥t♦ {X

g∈G

agδg :ag ∈Dg} ♠✉♥✐❞♦ ❞❛s ♦♣❡✲

r❛çõ❡s ❛❝✐♠❛ é ❞❡♥♦♠✐♥❛❞♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❛ss♦❝✐❛❞♦ ❛ α ❡

❞❡♥♦t❛❞♦ ♣♦rA⋊_αG✳

❖❜s❡r✈❛çã♦ ✶✳✷✳✷✳ ❱❡❥❛ q✉❡ ❛ ♠✉❧t✐♣❧✐❝❛çã♦ ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s✱ ❝♦♠♦Dg−1 é ✉♠ ✐❞❡❛❧ ❜✐❧❛t❡r❛❧ ❞❡A✱ ❡♥tã♦α_g−1(a_g)b_h∈D_g−1✳ ❖❜s❡r✈❛çã♦ ✶✳✷✳✸✳ ❆ ❛♣❧✐❝❛çã♦ ι : A → A⋊_{αG, a 7→ aδe} é ✉♠ ❤♦✲

♠♦♠♦r✜s♠♦ ❞❡ á❧❣❡❜r❛ ✐♥❥❡t✐✈♦✱ ❧♦❣♦ ♣♦❞❡♠♦s ✐❞❡♥t✐✜❝❛r A❝♦♠ ✉♠❛

s✉❜á❧❣❡❜r❛ ❞❡A⋊_αG✳

A⋊_αG é ✉♠❛ á❧❣❡❜r❛ ♥ã♦ ♥❡❝❡ss❛r✐❛♠❡♥t❡ ❛ss♦❝✐❛t✐✈❛✳ ❖✉ s❡❥❛✱ ❡♠

❣❡r❛❧✱ ♥ã♦ é ✈❡r❞❛❞❡ q✉❡(agδg·bhδh)·cjδj=agδg·(bhδh·cjδj)✳ ❚♦❞❛✈✐❛✱

❝♦♠♦ é ♠♦str❛❞♦ ❡♠ ❬✶❪✱ ✈❛❧❡ ♦ r❡s✉❧t❛❞♦ q✉❡ s❡❣✉❡ ❛s ❞❡✜♥✐çõ❡s ❛❜❛✐①♦✳

❉❡✜♥✐çã♦ ✶✳✷✳✹✳ ❙❡❥❛ A ✉♠❛ á❧❣❡❜r❛ ❡ s❡❥❛ I ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A✳ ❉✐③❡♠♦s q✉❡I é ✐❞❡♠♣♦t❡♥t❡ s❡I=I2_{✱ ✐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✉❡Ié ♥ã♦✲❞❡❣❡♥❡r❛❞♦ s❡ ♣❛r❛ ❝❛❞❛ ❡❧❡♠❡♥t♦ ♥ã♦✲♥✉❧♦a∈I

❡①✐st❡b∈I t❛❧ q✉❡ab6= 0♦✉ba6= 0✳

❈♦r♦❧ár✐♦ ✶✳✷✳✻✳ ❙❡❥❛ α ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ G ✉♠ ✉♠❛

á❧❣❡❜r❛ A t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ t ∈ G✱ t❡♠✲s❡ q✉❡ Dt é ✐❞❡♠♣♦t❡♥t❡ ♦✉

♥ã♦✲❞❡❣❡♥❡r❛❞♦✳ ❊♥tã♦ ♦ ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧A⋊_αGé ❛ss♦❝✐❛t✐✈♦✳

❖ r❡s✉❧t❛❞♦ ❛❝✐♠❛ ♥♦s ❞á ✉♠❛ ❝♦♥❞✐çã♦ s✉✜❝✐❡♥t❡ ♣❛r❛ q✉❡A⋊_αG

s❡❥❛ ❛ss♦❝✐❛t✐✈♦✱ s❡♥❞♦ q✉❡ t❛❧ ❝♦♥❞✐çã♦ ❞❡♣❡♥❞❡ ❞❡ ♣r♦♣r✐❡❞❛❞❡s ❞♦s ✐❞❡❛✐sDt✳ ❊♠ s❡❣✉✐❞❛✱ ♥♦s ♣r❡♦❝✉♣❛♠♦s ❝♦♠ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡A⋊αG✳

❈♦♥s✐❞❡r❛♥❞♦ ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ α = ({Dt}t∈G,{αt}t∈G) ❞❡ ✉♠

❣r✉♣♦ ❛❜❡❧✐❛♥♦ G❡♠ ✉♠ ❛♥❡❧R✱ ❡♠ ❬✹❪✱ ❡st✉❞❛♠♦s ❝♦♥❞✐çõ❡s ♥❡❝❡ssá✲

r✐❛s ❡ s✉✜❝✐❡♥t❡s ♣❛r❛ q✉❡A⋊_αGs❡❥❛ s✐♠♣❧❡s✳

❉✉r❛♥t❡ ♦s ♥♦ss♦s ❡st✉❞♦s ♣❛r❛ ♦ ❝❛s♦ ❡♠ q✉❡ ♦ ❣r✉♣♦Gé ❛❜❡❧✐❛♥♦

❡ ❝❛❞❛Dt ♣♦ss✉✐ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ✭❞❡✜♥✐r❡♠♦s ♦ q✉❡ ✐ss♦ s✐❣♥✐✜❝❛ ❧♦❣♦

❡♠ s❡❣✉✐❞❛✮✱ ❝♦♥s❡❣✉✐♠♦s ❡♥❢r❛q✉❡❝❡r ❛ ❝♦♥❞✐çã♦ ❡♥❝♦♥tr❛❞❛ ❡♠ ❬✹❪ q✉❡ ❣❛r❛♥t❡ ❛ s✐♠♣❧✐❝✐❞❛❞❡ ❞❡A⋊_αG✳

❆❣♦r❛ ❢❛❧❛r❡♠♦s s♦❜r❡ ♦s ♣ré✲r❡q✉✐s✐t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ❡♥✉♥❝✐❛r✲ ♠♦s ♦ t❡♦r❡♠❛ ♠❡♥❝✐♦♥❛❞♦ ❛❝✐♠❛✳ ◆♦ q✉❡ s❡❣✉❡✱Gé ✉♠ ❣r✉♣♦ ❛❜❡❧✐✲ ❛♥♦✱A é ✉♠ ❛♥❡❧ ❡ ❡stã♦ ♦♠✐t✐❞❛s ❛❧❣✉♠❛s ❞❡♠♦♥str❛çõ❡s q✉❡ ♣♦❞❡♠

s❡r ❡♥❝♦♥tr❛❞❛s ❡♠ ❬✹❪✳

❉❡✜♥✐çã♦ ✶✳✷✳✼✳ ❙❡❥❛a=X t∈G

atδt∈A⋊αG✳ ❉❡✜♥✐♠♦s✿

✭✐✮ ♦ s✉♣♦rt❡ ❞❡ a✱ ❞❡♥♦t❛❞♦ ♣♦rsupp(a)✱ ❝♦♠♦ ♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦

{t∈G:at6= 0};

✭✐✐✮ ❛ ♣r♦❥❡çã♦ ❞❡a ♥❛ ❝♦♦r❞❡♥❛❞❛g✱Pg:A⋊αG→A✱ ♣♦r

Pg(

X

t∈G

atδt) =ag.

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

❧♦❝❛✐s s❡ ♣❛r❛ t♦❞♦ ❝♦♥❥✉♥t♦ ✜♥✐t♦{r1, r2, . . . , rn} ⊆A❡①✐st❡e∈At❛❧

q✉❡e2_=e❡er

i =ri=rie , ♣❛r❛ t♦❞♦i∈ {1, . . . , n}✳

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

❥✉♥t♦ ❞❡ ✐❞❡♠♣♦t❡♥t❡s t❛✐s q✉❡ ♣❛r❛ t♦❞♦ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦{r1, . . . , rn}

❞❡A❡①✐st❡e∈E t❛❧ q✉❡eri =rie=ri✱ ♣❛r❛ t♦❞♦i∈ {1, . . . , n}✳

Pr♦♣♦s✐çã♦ ✶✳✷✳✶✵✳ ❙❡❥❛E⊆A✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A✳ ❊♥tã♦ Eδ0 ={eδ0 :e∈E} é ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛

A⋊_αG✳

❉❡♠♦♥str❛çã♦✿ ❙❡❥❛ {a1_{, . . . , a}n_{} ✉♠ s✉❜❝♦♥❥✉♥t♦ ✜♥✐t♦ ❞❡A⋊} αG✱

❡ ❡s❝r❡✈❡♠♦s ai ₌ X

t∈G

aitδt✱ ♣❛r❛ t♦❞♦ i ∈ {1, . . . , n}✳ ❊♥tã♦ ❡①✐st❡

s♦♠❡♥t❡ ✉♠❛ q✉❛♥t✐❞❛❞❡ ✜♥✐t❛ ❞❡ai

t✬s q✉❡ sã♦ ❞✐❢❡r❡♥t❡s ❞❡ ③❡r♦✳

❙❡❥❛X ={αt−1(ai_t) :ai_t6= 0} ∪ {ai_t:ai_t6= 0}✳ ❈♦♠♦X é ✜♥✐t♦ ❡A ♣♦ss✉✐ ✉♥✐❞❛❞❡s ❧♦❝❛✐s✱ s❡❥❛e∈E ✉♥✐❞❛❞❡ ❧♦❝❛❧ ♣❛r❛X✳ ❉❛í✱

eδ0aitδt=eaitδt=atiδt ❡aitδteδ0=αt(αt−1(ai_t)e)δ_t=α_t(α_t−1(ai_t))δ_t=ai_tδ_t. ❆❧é♠ ❞✐ss♦✱ t❡♠♦s q✉❡

(eδ0)2=eδ0·eδ0=α0(α0(e)·e)δ0=α0(e·e)δ0=α0(e)δ0=eδ0,

❝♦♠♦ q✉❡rí❛♠♦s✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✶✳ ❙❡❥❛ α= ({Dt}t∈G,{αt}t∈G)✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡

✉♠ ❣r✉♣♦G❡♠ ✉♠ ❛♥❡❧A✳ ❉✐③❡♠♦s q✉❡ ✉♠ ✐❞❡❛❧IEAé ●✲✐♥✈❛r✐❛♥t❡

s❡αg(I∩Dg−1)⊆I∩D_g✱ ♣❛r❛ t♦❞♦g∈G✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✷✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❛♥❡❧ A é ●✲s✐♠♣❧❡s s❡ ♦s ú♥✐❝♦s

✐❞❡❛✐s ●✲✐♥✈❛r✐❛♥t❡s ❞❡ Asã♦A❡{0}✳

▲❡♠❛ ✶✳✷✳✶✸✳ ❙❡❥❛ E ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A✳ ❙❡❥❛ α ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ ✭❛❞✐t✐✈♦✮ G t❛❧ q✉❡ Dt t❡♠

✉♥✐❞❛❞❡s ❧♦❝❛✐s✱ ♣❛r❛ t♦❞♦ t∈G✳ ❙✉♣♦♥❤❛ q✉❡ A é ●✲s✐♠♣❧❡s✳

❊♥tã♦✱ ♣❛r❛ t♦❞♦ ❡❧❡♠❡♥t♦ ♥ã♦ ♥✉❧♦r∈A⋊_αG✱ ❡ ♣❛r❛ ❝❛❞❛ ✉♥✐❞❛❞❡

❧♦❝❛❧ e∈E✱ ❡①✐st❡ r′∈A⋊_αG=:R t❛❧ q✉❡✿

❛✳ r′∈RrR❀

❜✳ P0(r′) =e❀

❝✳ # supp(r′_{)≤# supp(r)✳}

❉❡✜♥✐çã♦ ✶✳✷✳✶✹✳ ❙❡❥❛R✉♠ ❛♥❡❧✳ ❖ ❝❡♥tr♦ ❞❡Ré ♦ ❝♦♥❥✉♥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✉❡ra, b∈C(R)❡r∈Rt❡♠♦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✳

P❛r❛A⋊_G=:S❡_E⊆A✱ ❞❡♥♦t❛r❡♠♦s ♣♦r_Ce♦ ❝❡♥tr♦ ❞❡_eδ0Seδ0✳

❉❡✜♥✐çã♦ ✶✳✷✳✶✻✳ ❙❡❥❛♠R =A⋊_αG ♣r♦❞✉t♦ ❝r✉③❛❞♦ ♣❛r❝✐❛❧ ❡ _E

✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛A✳ P❛r❛ ❝❛❞❛e∈E✱ ❞❡✜♥✐♠♦s ♦

❝❡♥tr♦ ❞❡eδ0Reδ0 ❝♦♠♦

Ce:={x∈eδ0Reδ0:xy=yx, ♣❛r❛ t♦❞♦y∈eδ0Reδ0}.

▲❡♠❛ ✶✳✷✳✶✼✳ ❈♦♥s✐❞❡r❡ ❛s ♠❡s♠❛s ❝♦♥❞✐çõ❡s ❞♦ ❧❡♠❛ ❛♥t❡r✐♦r ❡ s❡❥❛

e∈E✳ ❊♥tã♦ t♦❞♦ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡ A⋊_αGt❡♠ ✐♥t❡rs❡çã♦ ♥ã♦✲♥✉❧❛

❝♦♠Ce∩ {eδ0+

X

g∈G\{0}

bgδg}✳

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

(i)❡(ii)sã♦ ❡q✉✐✈❛❧❡♥t❡s✳

❚❡♦r❡♠❛ ✶✳✷✳✶✽✳ ❙❡❥❛ E ✉♠ ❝♦♥❥✉♥t♦ ❞❡ ✉♥✐❞❛❞❡s ❧♦❝❛✐s ♣❛r❛ A ❡

s❡❥❛α✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ ✉♠ ❣r✉♣♦ ❛❜❡❧✐❛♥♦ G t❛❧ q✉❡ t♦❞♦ ✐❞❡❛❧Dt

t❡♠ ✉♥✐❞❛❞❡s ❧♦❝❛✐s✳ ❊♥tã♦ sã♦ ❡q✉✐✈❛❧❡♥t❡s✿ ✭✐✮ A⋊_αGé s✐♠♣❧❡s❀

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

✭✐✐✐✮ A é ●✲s✐♠♣❧❡s ❡Ce é ❝♦r♣♦✱ ♣❛r❛ ❛❧❣✉♠ e∈E✳

❉❡♠♦♥str❛çã♦✿ (ii)⇒(iii)➱ ó❜✈✐♦✳

(iii)⇒(i)❙❡❥❛J ✉♠ ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡A⋊_αG✳ P❡❧♦ ❧❡♠❛ ❛♥t❡r✐♦r✱

❡①✐st❡r∈(J∩Ce)\ {0}✳ ❈♦♠♦Ce é ❝♦r♣♦✱ ❡♥tã♦eδ0=r·r−1∈J✳

❈♦♥s✐❞❡r❡ ♦ ♠♦r✜s♠♦ϕ:A→A⋊_{αG a7→aδ0}✳ ❊♥tã♦ é ❢á❝✐❧ ✈❡r

q✉❡ϕ−1_{(J)é ✐❞❡❛❧ ♥ã♦✲♥✉❧♦ ❞❡A✳}

❱❛♠♦s ♠♦str❛r q✉❡ϕ−1(J)é ●✲✐♥✈❛r✐❛♥t❡✳ ❙❡❥❛a∈ϕ−1(J)∩D−h✳

❙❡❥❛eh✉♥✐❞❛❞❡ ♣❛r❛ a❡♠D−h✳ ❉❛í✱

αh(eh)δh·aδ0·ehδ−h=αh(α−h(αh(eh)a))δh·ehδ−h =αh(eha)δh·ehδ−h

=αh(a)δh·ehδ−h =αh(α0(a)eh)δ0

=αh(a)δ0.

▲♦❣♦ αh(a)δ0 ∈ J ❡ ❡♥tã♦ αh(a) ∈ ϕ−1(J)✱ ❞♦♥❞❡ ϕ−1(J) é G✲

✐♥✈❛r✐❛♥t❡ ❡ ϕ−1_{(J) =A✱ ♣♦✐sAéG✲s✐♠♣❧❡s ❡ϕ}−1_{(J)6= 0✳ P♦rt❛♥t♦✱}

aδ0∈J ♣❛r❛ t♦❞♦a∈A✳

❆❣♦r❛✱ s❡❥❛ag∈Dg q✉❛❧q✉❡r✳ ❙❡❥❛ eg ✉♥✐❞❛❞❡ ♣❛r❛ ag∈Dg✳ ❉❛í✱ agδ0·egδg=α0(α0(ag)eg)δg=α0(ageg)δg=agδg.

▲♦❣♦✱ agδg ∈ J, ♣❛r❛ t♦❞♦ag ∈ Dg, ♣❛r❛ t♦❞♦g ∈ G✳ ❉♦♥❞❡✱ J =A⋊_αG❡✱ ♣♦rt❛♥t♦✱_A⋊_αGé 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❛❞♦ ♣♦rE✳✳

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

✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦

❉❡✜♥✐çã♦ ✷✳✶✳✶✳ ❯♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦E= (E0_{, E}1_{, r, s)❝♦♥s✐st❡ ❞❡ ❝♦♥✲}

❥✉♥t♦s ♥ã♦ ✈❛③✐♦sE0_{, E}1_{❡ ❢✉♥çõ❡sr, s:E}1_→E0_✳

❖s ❡❧❡♠❡♥t♦s ❞❡ E0 _{sã♦ ❝❤❛♠❛❞♦s ✈ért✐❝❡s ❡ ♦s ❡❧❡♠❡♥t♦s ❞❡ E}1 _sã♦

❞❡♥♦♠✐♥❛❞♦s ❛r❡st❛s✳

P❛r❛ ✉♠❛ ❛r❡st❛e✱r(e)é ♦ r❛♥❣❡ ❞❡ e❡s(e)é ♦ s♦✉r❝❡ ❞❡e✳

❊①❡♠♣❧♦ ✷✳✶✳✷✳ ❙❡❥❛E={E0_{, E}1_{, r, s}✱ ❡♠ q✉❡E}0_{={u, v, w, k}, E}1₌

{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

u f //v g w e O O k h o o

❊①❡♠♣❧♦ ✷✳✶✳✸✳ ❙❡❥❛ E={E0_{, E}1_{, r, s}✱ ❡♠ q✉❡E}0_{={u, v, w, k, z}✱}

E1 _={f

1, f2, f3, f4, f5, f6, f7}✱z =s(f7)✱u=r(f1) =s(f1) =s(f2) =

s(f3)✱ v = r(f3) =s(f4) = s(f6)✱ w = r(f4) = s(f5) ❡k = r(f2) =

r(f6) =r(f5) =r(f7)✳ P♦❞❡♠♦s r❡♣r❡s❡♥t❛r ❡st❡ ❣r❛❢♦ ♣♦r

u f1 9 9 f3 / / f2 v f6 f4 / /w f5   k z f7 o o

❉❡✜♥✐çã♦ ✷✳✶✳✹✳ ❙❡❥❛ E ✉♠ ❣r❛❢♦ ❞✐r✐❣✐❞♦ ❡ s❡❥❛ K ✉♠ ❝♦r♣♦✳ ❆

á❧❣❡❜r❛ ❞❡ ▲❡❛✈✐tt ❞❡ E ❝♦♠ ❝♦❡✜❝✐❡♥t❡s ❡♠ K✱ ❞❡♥♦t❛❞❛ ♣♦rLK(E)✱

é ❛ K✲á❧❣❡❜r❛ ✉♥✐✈❡rs❛❧ ❣❡r❛❞❛ ♣♦r ✉♠ ❝♦♥❥✉♥t♦ {v : v ∈ E0_{} ❞❡}

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

❡❧❡♠❡♥t♦s s❛t✐s❢❛③❡♥❞♦✿

✭✐✮ s(e)e=er(e) =e✱ ♣❛r❛ t♦❞♦e∈E1❀

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

✭✐✐✐✮ ♣❛r❛ q✉❛✐sq✉❡re, f∈E1_{t❡♠✲s❡ q✉❡e}∗_{f = 0s❡e6=f❡e}∗_e=r(e)❀ ✭✐✈✮ v= X

e∈E1_:s(e)=v

ee∗✱ ♣❛r❛ t♦❞♦ ✈ért✐❝❡v t❛❧ q✉❡0<#{e:s(e) =

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 ✉♠❛

s❡q✉ê♥❝✐❛ µ = µ1µ2. . . µn t❛❧ q✉❡ r(µi) = s(µi+1)✱ ♣❛r❛ t♦❞♦ i ∈

{1, . . . , n−1}✳ ❊s❝r❡✈❡♠♦s|µ|=n♣❛r❛ ♦ ❝♦♠♣r✐♠❡♥t♦ ❞❡µ✳ ❈❤❛♠❛✲

♠♦s ♦s ✈ért✐❝❡s ❞❡ ❝❛♠✐♥❤♦s ❞❡ t❛♠❛♥❤♦ ③❡r♦✳

❉❡✜♥✐çã♦ ✷✳✶✳✻✳ En _{é ♦ ❝♦♥❥✉♥t♦ ❞❡ ❝❛♠✐♥❤♦s ❞❡ t❛♠❛♥❤♦n✳}

❊st❡♥❞❡♠♦s ❛s ❛♣❧✐❝❛çõ❡s r❛♥❣❡ ❡ s♦✉r❝❡ ♣❛r❛ En _{❞❡✜♥✐♥❞♦s(µ) =} s(µ1)❡r(µ) =r(µ1)✱ s❡n≥2✱ ❡s(v) =v=r(v)✱ ♣❛r❛n= 0✳

❉❡✜♥✐çã♦ ✷✳✶✳✼✳ W é ♦ ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ✜♥✐t♦s ❡♠E✱

♦✉ s❡❥❛✱

W =

∞ [

n=1

{µ1. . . µn:µi∈E1 ❡r(µi) =s(µi+1), ∀i∈ {1, . . . , n−1}}.

❉❡✜♥✐çã♦ ✷✳✶✳✽✳ W∞_{é ❝♦♥❥✉♥t♦ ❞❡ t♦❞♦s ♦s ❝❛♠✐♥❤♦s ✐♥✜♥✐t♦s ❞❡E✱} ♦✉ s❡❥❛✱

W∞={µ1µ2. . .:µi∈E1❡r(µi) =s(µi+1), ∀i∈N}.

❆❣♦r❛ ✈❛♠♦s ❡st❡♥❞❡rs, r:E1_→E0 _{♣❛r❛W∪W}∞_∪E0_✳

❉❡✜♥❛✿

s(µ) =s(µ1)✱ ♣❛r❛µ=µ1µ2. . .∈W∞ ♦✉µ=µ1µ2. . . µn❀ r(ξ) =r(ξn)✱ ♣❛r❛ξ=ξ1. . . ξn∈W❀

r(v) =v=s(v)✱ ♣❛r❛v∈E0✳

❉❡✜♥✐çã♦ ✷✳✶✳✾✳ ❉✐③❡♠♦s q✉❡ ✉♠ ❝❛♠✐♥❤♦ξ=ξ1. . . ξné ♦ ❝♦♠❡ç♦ ❞♦

❝❛♠✐♥❤♦η=η1. . . ηm s❡m≥n❡ηi=ξi, ♣❛r❛ t♦❞♦i∈ {1, . . . , n}✳

❉❡✜♥✐çã♦ ✷✳✶✳✶✵✳ ❖ ❣r❛✉ ❞❡ s❛í❞❛ ❞❡ ✉♠ ✈ért✐❝❡ v é ♦ ♥ú♠❡r♦ ❞❡

❛r❡st❛set❛✐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♦✳

❙❡❥❛F♦ ❣r✉♣♦ ❧✐✈r❡ ❣❡r❛❞♦ ♣♦r_E1❡ s❡❥❛_{X={ξ∈W :r(ξ)}é ✉♠ ♣♦ç♦_}∪

{v∈E0_{:v é ✉♠ ♣♦ç♦} ∪W}∞_✳ P❛r❛ ❝❛❞❛ c∈F✱ ❞❡✜♥❛✿

X0=X, ❡♠ q✉❡0 é ♦ ❡❧❡♠❡♥t♦ ♥❡✉tr♦ ❞❡F❀

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

Xa={ξ∈X:ξ1ξ2. . . ξ|a|=a}✱ ♣❛r❛ t♦❞♦a∈W❀

Xab−1 = {ξ ∈ X : ξ₁ξ₂. . . ξ_|a| = a} = X_a✱ ♣❛r❛ ab−1 ∈ F✱ ❝♦♠ a, b ∈ W , r(a) = r(b)❡ab−1 _{♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ ♦✉ s❡❥❛✱}

a|a|6=b|b|✱

Xc=∅✱ ♣❛r❛ q✉❛❧q✉❡r ♦✉tr♦c∈F✳

❖❜s❡r✈❛çã♦ ✷✳✶✳✶✷✳ P♦❞❡♠♦s ✈❡rW ❝♦♠♦ ✉♠ s✉❜❝♦♥❥✉♥t♦ ❞❡F✳

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✸✳ r(b) ∈ Xb−1 s❡✱ ❡ s♦♠❡♥t❡ s❡✱ r(b) é ✉♠ ♣♦ç♦✳ ❆❧é♠ ❞✐ss♦✱ s❡ r(b)é ✉♠ ♣♦ç♦ ❡♥tã♦Xb−1 ={r(b)} ❡X_b={b}✳ ❉❡♠♦♥str❛çã♦✿ ❙✉♣♦♥❤❛ q✉❡ r(b)∈ Xb−1✳ ❊♥tã♦ r(b) ∈X ∩E0 ❡✱ ♣❡❧❛ ❞❡✜♥✐çã♦ ❞❡X✱ s❡❣✉❡ q✉❡ r(b)é ♣♦ç♦✳ ❙✉♣♦♥❤❛ ❛❣♦r❛ q✉❡r(b)é

♣♦ç♦✳ ❈♦♠♦r(b)∈X ❡r(b) =s(r(e))✱ t❡♠♦s q✉❡r(b)∈Xb−1✳

❙❡r(b) é ♣♦ç♦✱ ❡♥tã♦✱ ♣❛r❛ξ ∈ Xb−1✱ r(b) = s(ξ) ✐♠♣❧✐❝❛ ❡♠ ξ ∈

E0_{∩X✳ ❉♦♥❞❡ ξ = r(b)✳ P❛r❛ ξ ∈ X}

b t❡♠♦s ξ1. . . ξ|b| = b✱ ♦ q✉❡ ✐♠♣❧✐❝❛ ❡♠ r(ξ|b|) =r(b)s❡r ♣♦ç♦✳ ▲♦❣♦✱ξ=ξ1. . . ξ|b|=b✳

❖❜s❡r✈❛çã♦ ✷✳✶✳✶✹✳ P❛r❛ t♦❞♦ v ∈ E0_{✱ t❡♠♦s v ∈ X}

v := {ξ ∈ X : s(ξ) =v}s❡✱ ❡ s♦♠❡♥t❡ s❡✱v é ✉♠ ♣♦ç♦✳ ❉❡ ❢❛t♦✱

v∈Xv ⇐⇒ v∈X ⇐⇒ v é ♣♦ç♦.

❆❧é♠ ❞✐ss♦✱ s❡v é ✉♠ ♣♦ç♦✱ ❡♥tã♦Xv={v}✳

▲❡♠❛ ✷✳✶✳✶✺✳ ❙❡❥❛♠ a, c∈W , b, d∈W ∪ {0} ❡v∈E0_{✳ ❊♥tã♦✿}

(1)Xa−1∩X_c−1= (

Xa−1 =X_c−1 ✱ s❡ r(a) =r(c)

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(2)Xa−1∩X_cd−1 = (

Xcd−1 ✱ s❡ r(a) =s(c)

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(3) ❙✉♣♦♥❤❛r(a) =r(b)❡r(c) =r(d)✳

Xab−1∩X_cd−1 =   

 

Xab−1 ✱ s❡a=ct, ♣❛r❛ ❛❧❣✉♠t∈W ∪ {0}

Xcd−1 ✱ s❡c=at, ♣❛r❛ ❛❧❣✉♠t∈W ∪ {0}

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(4)Xv∩Xb−1= (

Xv =Xb−1 ✱ s❡r(b) =v

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(5)Xv∩Xab−1 = (

Xab−1 ✱ s❡s(a) =v

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(6) Xv =

[

s(a)=v Xab−1

❉❡♠♦♥str❛çã♦✿

(1)Xa−1∩X_c−1 ={ξ∈X :s(ξ) =r(a)} ∩ {η ∈X:s(η) =r(c)}

={ξ∈X :s(ξ) =r(a) =r(c)}

❊♥tã♦

Xa−1∩X_c−1= (

Xa−1 =X_c−1 ✱ s❡r(a) =r(c)

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(2) ❙❡r(c) =r(d)❡cd−1_{❡stá ♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ ❡♥tã♦✿}

Xa−1∩X_cd−1 ={ξ∈X :s(ξ) =r(a)} ∩ {η ∈X:η₁η₂. . . η_|c|=c}

={ξ∈X :s(ξ) =r(a)❡η1η2. . . η|c|=c}

={ξ∈X :s(ξ) =r(a) =s(c)❡cé ❝♦♠❡ç♦ ♣❛r❛ ♦ ❝❛♠✐♥❤♦ξ}

❙❡r(c)6=r(d)✱ ❡♥tã♦ Xcd−1 =∅✳ ▲♦❣♦✱

Xa−1∩X_cd−1 = (

Xcd−1 =X_c−1 ✱ s❡r(a) =s(c)

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(3)Xab−1∩X_cd−1 =X_a∩X_c

={ξ∈X :ξ1. . . x|a|=a} ∩ {η∈X :η1. . . η|c|=c}

={ξ∈X :aé ❝♦♠❡ç♦ ❞❡ξ❡cé ❝♦♠❡ç♦ ❞❡ξ}

❚❡♠♦s três ❝❛s♦s✿

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

❝❛s♦ ✷✿ c é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦a✳

◆❡st❡ ❝❛s♦✱ ❡①✐st❡t∈W∪ {0} t❛❧ q✉❡a=ct✳ ❉❛í✱

Xab−1∩X_cd−1=X_a∩X_c={ξ∈X :aé ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ξ}=X_ab−1. ❝❛s♦ ✸✿ aé ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ c✳

◆❡st❡ ❝❛s♦✱ ❡①✐st❡t∈W∪ {0} t❛❧ q✉❡c=at✳ ❉❛í✱

Xb−1∩X_cd−1 =X_a∩X_c={ξ∈X:cé ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ξ}=X_cd−1.

(4)Xv∩Xb−1 ={ξ∈X:s(ξ) =r(v) =v} ∩ {η∈X :s(η) =r(b)}

={ξ∈X:s(ξ) =v=r(b)}.

❊♥tã♦✱

Xv∩Xb−1 = (

Xv=Xb−1 ✱ s❡r(b) =v,

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

✭✺✮ ❙❡r(a)6=r(b)✱ ❡♥tã♦ Xab−1 =∅✳ ❙❡r(a) =r(b)✱ ❡♥tã♦✿

Xv∩Xab−1={ξ∈X :s(ξ) =v} ∩ {η∈X :aé ❝♦♠❡ç♦ ❞❡η}

={ξ∈X :s(ξ) =v=s(a)❡aé ❝♦♠❡ç♦ ❞❡ξ}.

▲♦❣♦✱

Xv∩Xab−1 = (

Xab−1 ✱ s❡s(a) =v

∅ ✱ ❝❛s♦ ❝♦♥trár✐♦

(6) ❙❡❥❛ξ∈Xv✱ ♦✉ s❡❥❛✱s(ξ) =v✳ ❙❡ξ∈W∩X✱ ❡♥tã♦ξ∈Xη ⊆

[

s(a)=v

Xab−1✳ ❙❡ ξ ∈ {u∈E0 :✉ é ♣♦ç♦}✱ ❡♥tã♦ s(ξ) = r(ξ) =v✳ ❙❡

ξ∈W∞_{✱ ❡♥tã♦s(ξ) =s(ξ}

1) =v❡✱ s❡♥❞♦ ❛ss✐♠✱ ξ∈Xξ1 [

s(a)=v Xab−1✳ ❆ ♦✉tr❛ ✐♥❝❧✉sã♦ s❡❣✉❡ ❞♦ ✐t❡♠ ✭✺✮✳

❚❡♠♦s ❛té ❛❣♦r❛ ♦ ❣r✉♣♦ ❧✐✈r❡ F ❣❡r❛❞♦ ♣♦r _E1 ❡ ♦ ❝♦♥❥✉♥t♦ _X

s♦❜r❡ ♦ q✉❛❧F✐rá ❛❣✐r✳ ❆✐♥❞❛ ❢❛❧t❛ ❞❡✜♥✐r♠♦s✱ ♣❛r❛ t♦❞♦_c∈Ft❛❧ q✉❡

Xc−1 6=∅✱ ❜✐❥❡çõ❡sθ_c:X_c−1 →X_c✳

❉❡✜♥❛ θ0 : X0 → X0 ❝♦♠♦ ❛ ❛♣❧✐❝❛çã♦ ✐❞❡♥t✐❞❛❞❡✳ P❛r❛ b ∈ W✱

❞❡✜♥❛ θb :Xb−1 →X_b , ♣♦r ξ 7→bξ✳ ◆♦t❡ q✉❡ θ_b ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s✱ ♣❛r❛ ξ∈Xb−1 t❡♠♦s q✉❡s(ξ) =r(b)❡✱ ♣♦rt❛♥t♦✱bξ∈X_b✳

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

Xab−1 , ♣♦rξ 7→ aξ_|b|+1ξ_|b|+2. . .✳ ◆♦t❡ q✉❡ θ_ab−1 ❡stá ❜❡♠ ❞❡✜♥✐❞❛✱ ♣♦✐s✱ ♣❛r❛ ξ∈ Xba−1✱ t❡♠♦s q✉❡ ξ₁. . . ξ_|b| =b ❡✱ ❝♦♠♦ r(a) =r(b) =

r(ξ|b|) =s(ξ|b|+1)✱ s❡❣✉❡ q✉❡aξ|b|+1ξ|b|+2. . .∈Xab−1✳

❆✐♥❞❛✱ ❞❡✜♥❛θb−1 : X_b → X_b−1✱ ♣♦r η 7→η_|b|+1η_|b|+2. . .✱ s❡r(b) ♥ã♦ é ♣♦ç♦✱ ❡ η7→r(b)✱ s❡r(b)é ♣♦ç♦✳

P♦r ✜♠✱ ❞❡✜♥❛θba−1 :X_ab−1→X_ba−1✱ ♣♦r η 7→bη_|a|+1η_|a|+2. . .✳ Pr♦♣♦s✐çã♦ ✷✳✶✳✶✻✳ P❛r❛ b ∈ W✱ t❡♠♦s θb−1 = θ−_b1✳ P❛r❛ a, b ∈

W ❝♦♠r(a) =r(b)❡ a|a|=6 b|b|✱ t❡♠♦sθ_ab−1−1 =θba−1✳ ❉❡♠♦♥str❛çã♦✿ ❉❛❞♦η∈Xb−1✱ s❡❣✉❡ q✉❡

(θb−1◦θ_b)(η) =θ−1

b (η|b|+1η|b|+2. . .) =bη|b|+1η|b|+2. . .=η,

s❡r(b)♥ã♦ é ♣♦ç♦✳ ❙❡ r(b)é ♣♦ç♦✱ s❡❣✉❡ q✉❡

(θb−1◦θ_b)(η) =θ_b−1(r(b)) =br(b) =η.

❉❛❞♦ξ∈Xb✱ s❡❣✉❡ q✉❡(θb◦θb−1)(ξ) =θ_b(bξ) =ξ✳ P❛r❛ξ∈Xba−1✱ s❡❣✉❡ q✉❡

(θba−1◦θ_ab−1)(ξ) = (θ_ba−1)(aξ_|b|+1ξ_|b|+2. . .) =bξ_|b|+1ξ_|b|+2. . .=ξ.

P❛r❛η∈Xab−1 s❡❣✉❡ q✉❡

(θab−1◦θ_ba−1)(η) = (θ_ab−1)(bη_|a|+1η_|a|+2. . .) =aη_|a|+1η_|a|+2. . .=η.

Pr♦♣♦s✐çã♦ ✷✳✶✳✶✼✳ ❖ ♣❛rθ= ({Xc}c∈F,{θc}c∈F)é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳ ❉❡♠♦♥str❛çã♦✿ ❖ ❛①✐♦♠❛ ✭✐✮ ❞❡ ✶✳✶✳✶ s❡❣✉❡ ❞❛s ❞❡✜♥✐çõ❡s ❞❡X0❡θ0✳

P❛r❛ ♦s ♦✉tr♦s ❛①✐♦♠❛s ❞❡ ✶✳✶✳✶✱ ❜❛st❛ q✉❡ ✈❡r✐✜q✉❡♠♦s ♣❛r❛g, h∈ {ab−1_{:a, b∈W ∪ {0}} ∪ {0}✱ ♣♦✐s✱ ❝❛s♦ ❝♦♥trár✐♦✱X}

g∩Xh−1 =∅❡ ❛ ✐♥❝❧✉sã♦ ♦✉ ✐❣✉❛❧❞❛❞❡ é tr✐✈✐❛❧✳

◆♦t❡ q✉❡{ab−1_{:a, b∈W∪ {0}} ∪ {0}é ❢❡❝❤❛❞♦ ♣♦r ✐♥✈❡rs♦s✳ ◆♦t❡}

❛✐♥❞❛ 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 ∪}

{0}✱ ab−1_{, cd}−1 _{♥❛ ❢♦r♠❛ r❡❞✉③✐❞❛✱ r(a) = r(b)✱ r(c) = r(d)✱ a, b ♥ã♦}

s✐♠✉❧t❛♥❡❛♠❡♥t❡0 ❡ ♥❡♠c, ds✐♠✉❧t❛♥❡❛♠❡♥t❡0✳

✭✐✐✮ ❱❛♠♦s ♠♦str❛r q✉❡θ_h−1(Xh∩Xg−1)⊆X_(gh)−1✳ ❚❡♠♦s ✸ ❝❛s♦s✳ ❝❛s♦ ✶✿ a♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦d❡d♥ã♦ é ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦

a✳

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

Xg∩Xh−1=X_ab−1∩X_(cd−1₎−1 =X_a∩X_d=∅, ♣♦✐s ♥❡♥❤✉♠ξ∈X ♣♦❞❡ ❝♦♠❡ç❛r ❝♦♠a❡ds✐♠✉❧t❛♥❡❛♠❡♥t❡✳

❝❛s♦ ✷✿ aé ❝♦♠❡ç♦ ❞♦ ❝❛♠✐♥❤♦ d✱ ❡ ❡s❝r❡✈❡♠♦s d=ad′_✳ ◆♦t❡ q✉❡a6= 0⇒d6= 0✳ ❉❛í✱

Xg∩Xh−1 =X_ab−1∩X_(cd−1₎−1

=     

Xb−1∩X_c−1 ✱ s❡a=d= 0

Xb−1∩X_d ✱ s❡a= 0❡d6= 0

Xa∩Xd ✱ s❡a6= 0

=     

Xb−1∩X_c−1 ✱ s❡a=d= 0

Xb−1∩X_d ✱ s❡a= 0❡d6= 0

Xd ✱ s❡a6= 0

❊♥tã♦✱

θg−1(Xg∩Xh−1) =θba−1(X_g∩X_h−1)

=     

θb(Xb−1∩X_c−1) ✱ s❡a=d= 0✭✯✮

θb(Xb−1∩X_d) ✱ s❡a=d❡d6= 0✭✯✯✮

θba−1(X_d) ✱ s❡a6= 0✭✯✯✯✮

❊♠(∗)✱ t❡♠♦s q✉❡ g−1h−1=ba−1dc−1=bc−1❡b6= 0✱ ♣♦✐s a= 0✳

❊♥tã♦✱

Xg−1_h−1 =X_bc−1 =X_b. ▲♦❣♦✱

θg−1(Xg∩Xh−1) =θ_b(X_b−1∩X_c−1)⊆X_b =X_g−1_h−1,

❝♦♠♦ q✉❡rí❛♠♦s✳

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

▲♦❣♦✱bd6= 0❡ ❡♥tã♦

Xg−1_h−1 =X_bdc−1 =X_bd.

❉❛í✱ ♣❛r❛ t♦❞♦ξ∈Xb−1∩X_d t❡♠♦sξ=dξ′ ❡ t❛♠❜é♠

θg−1(ξ) =θ_ba−1(ξ) =θ_b(ξ) =bdξ′ = (bd)ξ′.

▲♦❣♦✱θb(ξ)∈Xbd✳ ❆ss✐♠✱

θg−1(X_g∩X_h−1) =θ_b(X_b−1∩X_d)⊆X_bd=X_g−1_h−1,

❝♦♠♦ q✉❡rí❛♠♦s✳

❊♠(∗∗∗)✱ t❡♠♦s ♠❛✐s ❞♦✐s ❝❛s♦s✳ ❙❡b=d′ _{= 0✱ ❡♥tã♦a=d❡r(c) =}

r(d) =r(a)✳ ▲♦❣♦✱Xc−1 =X_a−1✳ ❈♦♠♦X_g−1_h−1 =X_ba−1_dc−1 =X_c−1✱ ❡♥tã♦

θg−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✱ ❡♥tã♦}

Xg−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✉❡θg−1(X_d)⊆X_bd′✳ ❉❡ ❢❛t♦✱ ♣❛r❛ t♦❞♦ ξ∈X_d✱

❡s❝r❡✈❡♠♦sξ=dξ′_=ad′_ξ′ _{❡ ❡♥tã♦}

θba−1(ξ) =θ_ba−1(ad′ξ′) =bd′ξ∈X_bd′.

❝❛s♦ ✸✿ dé ❝♦♠❡ç♦ ❞❛ ♣❛❧❛✈r❛a✱ ❡ ❡s❝r❡✈❡♠♦sa=da′✳

❊st❡ ❝❛s♦ é s✐♠✐❧❛r ❛♦ ❝❛s♦ ❛♥t❡r✐♦r✳

✭✐✐✐✮ θg◦θh(ξ) =θgh(ξ)✱ ♣❛r❛ t♦❞♦ξ∈θh−1(X_h∩X_g−1)✳ ❚❡♠♦s ✸ ❝❛s♦s✳

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

◆❡st❡ ❝❛s♦✱Xg∩Xh−1=∅✱ ❝♦♠♦ ♥♦ ♣r✐♠❡✐r♦ ❝❛s♦ ❞♦ ❛①✐♦♠❛ ✭✐✐✮✳ ❝❛s♦ ✷✿ aé ❝♦♠❡ç♦ ❞❛ ♣❛❧❛✈r❛d✱ ❡ ❡s❝r❡✈❡♠♦sd=ad′_✳

❙❡❥❛ ξ ∈ θ−1

g (Xg∩Xh−1) = θ_ba−1(X_ab−1 ∩X_dc−1)✳ ❊♥tã♦ ξ =

θba−1(η)✱ ♣❛r❛ ❛❧❣✉♠ η ∈ X_ab−1 ∩X_dc−1✳ ❆ss✐♠✱ η = dη′ = ad′η′✳ ▲♦❣♦✱

ξ=θba−1(η) =θ_ba−1(ad′ξ′) =bd′η′. ❉❛í✱ ♣♦r ✉♠ ❧❛❞♦✱ t❡♠♦s q✉❡

θh(θg(ξ)) =θcd−1(θ_ab−1(bd′η′)) =θ_cd−1(ad′η′) =θ_cd−1(dη′) =cη′ P♦r ♦✉tr♦ ❧❛❞♦✱ s❡❣✉❡ q✉❡

θhg(ξ) =θc(bd′)−1((bd′)η′) =cη′.

❝❛s♦ ✸✿ dé ❝♦♠❡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✱ ❝♦♠_{Xc =6} ∅✱ s❡❥❛ F_(Xc) ❛

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

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

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

❝❛❞❛c∈F✱ ❞❡✜♥❛

αc: F(Xc−1) → F(X_c)

f 7→ f◦θc−1✳

P❛r❛ t♦❞♦c ∈F✱ _αc é ✉♠_K✲✐s♦♠♦r✜s♠♦✳ ❊✱ ♣❡❧♦ ❊①❡♠♣❧♦ ✶✳✶✳✻✱

s❡❣✉❡ q✉❡ α = ({F_{(Xc)}c∈F,{αc}c∈F)} é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧ ❞❡ F s♦❜r❡

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❛✳

P❛r❛ ✐ss♦✱ s❡❥❛1c∈F(Xc)❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠Xc✱ ♦✉ s❡❥❛✱

1c(ξ) = 1Xc(ξ) = [ξ∈Xc] =

(

1 ✱ s❡ξ∈Xc 0 ✱ s❡ξ /∈Xc

P❛r❛ ❝❛❞❛v∈E0_{✱ s❡❥❛1}

v∈F(Xv)❛ ❢✉♥çã♦ ❝❛r❛❝t❡ríst✐❝❛ ❡♠Xv✳

▲❡♠❛ ✷✳✶✳✶✽✳ ❙❡❥❛♠ p, q∈F✳ ❊♥tã♦✿

✭✐✮ αp(1p−11_q) = 1_p1_pq❀

✭✐✐✮ P❛r❛a∈W, b∈W ∪ {0}✱ s❡❣✉❡ q✉❡

αa(1a−11_v) = (

1a ✱ s❡r(a) =v 0 ✱ s❡r(a)6=v

❡

αab−1(1_ba−11_v) = (

1ab−1 ✱ s❡s(b) =v

0 ✱ s❡s(b)6=v

❉❡♠♦♥str❛çã♦✿ ✭✐✮ ❙✉♣♦♥❤❛p=ab−1 _❡q=cd−1_{✱ ❝♦♠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♦✿

αp(1p−11_q)(ξ) = (α_p(1_p−1)α_p(1_q))(ξ)

=αp(1p−1)(ξ)α_p(1_q)(ξ)

= 1p−1(θ_p−1(ξ))1_q(θ_p−1(ξ))

= 1ab−1(θ_ab−1(ξ))1_atd−1(θ_ab−1(ξ))

= [θab−1 ∈X_ab−1][θ_ab−1(ξ)∈X_atd−1]

= [ξ∈Xba−1][θ_ab−1(ξ)∈X_atd−1]

= [ξ∈Xbtd−1]

1p1pq(ξ) = 1p(ξ)1pq(ξ) = [ξ∈Xp][ξ∈Xpq]

= [ξ∈Xba−1∩X_ba−1_cd−1] = [ξ∈X_ba−1∩X_btd−1]

= [ξ∈Xbtd−1]

❆♥❛❧♦❣❛♠❡♥t❡✱ s❡ a = ct✱ ♣❛r❛ ❛❧❣✉♠ t ∈ W✱ ♠♦str❛✲s❡ q✉❡ ♦

r❡s✉❧t❛❞♦ s❡❣✉❡✳

❙❡a6=ct✱ ♣❛r❛ t♦❞♦t∈W✱ ❡c6=at✱ ♣❛r❛ t♦❞♦t∈W✱ ❡♥tã♦✿

1ab−11_cd−1 = 0✱ ♣❡❧♦ ❧❡♠❛ ✸✳✶✳✶ ❡

1ba−1_cd−1= 0 ✱ ♣❡❧❛ ❞❡✜♥✐çã♦✳ ❆s ♦✉tr❛s ♣♦ss✐❜✐❧✐❞❛❞❡s ♣❛r❛ p, q∈Fsã♦ s✐♠✐❧❛r❡s✳

✭✐✐✮ ❙❡❥❛♠a∈W, b∈W∪ {0}✳ P❛r❛ t♦❞♦ξ∈X✱ t❡♠♦s✿ αa(1a−11_v)(ξ) =α_a(1_a−1)(ξ)α_a(1_v)(ξ)

= 1a−1(θ_a−1)(ξ)1_v(θ_a−1)(ξ)

= [θa−1(ξ)∈X_a−1][θ_a−1(ξ)∈X_v]

= [ξ∈Xa][ξ∈Xv∩Xa] = [ξ∈Xv∩Xa]

=

(

1a ✱ s❡Xv=Xa 0 ✱ s❡r(a)6=v

αab−1(1_ab−11_v)(ξ) =α_ab−1(1_ab−1)(ξ)α_ab−1(1_v)(ξ)

= 1ba−1(θ_ba−1)(ξ)1_v(θ_ba−1)(ξ)

= [θba−1(ξ)∈X_ba−1][θ_ba−1(ξ)∈X_v]

= [ξ∈Xab−1][ξ∈X_ab−1∩X_v]

=

(

1ab−1 ✱ s❡X_v∩X_ab−1=X_ab−1 ✭ s❡s(a) =v)

0 ✱ 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✳

❙❡❥❛D(X) = D0 = span {1p:p∈F\ {0}} ∪ {1v:v∈E0} ✭s♣❛♥

❑✲❧✐♥❡❛r✮✳

❙❡❥❛✱ ♣❛r❛ ❝❛❞❛p∈F_{\ {0}}✱_Dp⊆F_(Xp)❞❡✜♥✐❞♦ ❝♦♠♦

Dp= 1pD0= span{1p1q :q∈F}.

❖❜s❡r✈❛çã♦ ✷✳✶✳✶✾✳ ◆♦t❡ q✉❡D(X)❡Dpsã♦ ❑✲á❧❣❡❜r❛s ❡ q✉❡Dpé ✉♠

✐❞❡❛❧ ❞❡D0✱ ♣❛r❛ t♦❞♦p∈F✳

❈♦♠♦ αp(1p−11_q) = 1_p1_pq✱ ❝♦♥s✐❞❡r❡✱ ♣❛r❛ ❝❛❞❛ p∈ F✱ ❛ r❡str✐çã♦ ❞❡αp ❛Dp−1✿

αp: Dp−1 → D_p

1p−11_q 7→ α_p(1_p−11_q) = 1_p1_pq ❊♥tã♦α= ({αp}p∈F,{Dp}p∈F)é ✉♠❛ ❛çã♦ ♣❛r❝✐❛❧✳ ❙❡❥❛D(X)⋊_αF ♦ ♣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❝✐❛❧✳

Pr♦♣♦s✐çã♦ ✷✳✷✳✶✳ ❊①✐st❡ ✉♠K✲❤♦♠♦♠♦r✜s♠♦ϕ❞❡LK(E)❡♠D(X)⋊α

F t❛❧ q✉❡✱ ♣❛r❛ t♦❞♦ _{e ∈ E}1✱ _{ϕ(e) = 1eδe} ❡_ϕ(e∗_{) = 1e}−1δ_e−1 ❡✱ ♣❛r❛ t♦❞♦ v∈E0✱ϕ(v) = 1vδ0✳

❉❡♠♦♥str❛çã♦✿ ❈♦♥s✐❞❡r❡ ♦s ❝♦♥❥✉♥t♦s{1eδe,1e−1δ_e−1 :e∈E1} ❡{1_vδ₀:

v∈E0_{}❡♠D(X)⋊}

αF✳

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

✭✐✮ 1s(e)δ01eδe=α0(α0(1s(e))1e)δe= 1s(e)1eδe= 1eδe✱ ♣♦✐s

1s(e)1eδe(ξ) = 1Xs(e)∩Xe(ξ) = [ξ∈Xs(e)∩Xe] = [ξ∈Xe] = 1e(ξ).

1eδe1r(e)δ0=αe(αe−1(1_e)1_r(e))δ_e=α_e(1_e−11_r(e))δ_e

= 1e1er(e)δe= 1e1eδe= 1eδe