P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✺
❈❉❉✿ ✺✶✶✳✸
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊
❊❙❚❘❯❈❚❯❘❆❙ ✭❉❊❙P❯➱❙ ❉❊ ◆✳❉❆ ❈❖❙❚❆ ✮✳
❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛ ② ❈s✳ ❞❡ ❧❛ ❈♦♠♣✉t❛❝✐ó♥ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ❙❛♥t✐❛❣♦ ❞❡ ❈❤✐❧❡✱ ▲❛s ❙♦♣❤♦r❛s ★ ✶✼✺✳ ❙❛♥t✐❛❣♦✱ ❈❤✐❧❡✳
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ❈✐❡♥❝✐❛s ❞❡ ❧❛ ❈♦♠♣✉t❛❝✐ó♥✱ ❇❡✉❝❤❡❢ ★ ✽✺✶✳ ❯♥✐✈❡rs✐❞❛❞ ❞❡ ❈❤✐❧❡✶✱
❙❛♥t✐❛❣♦✱ ❈❤✐❧❡✳
✈✐❞❛❧✳♥❛✈❛rr♦❅✉s❛❝❤✳❝❧
❘❡❝❡✐✈❡❞✿ ✷✷✳✵✹✳✷✵✶✺❀ ❆❝❝❡♣t❡❞✿ ✷✹✳✵✺✳✷✵✶✺
❆❜str❛❝t✿ ❚❤❡ ♣r❡s❡♥t ❛rt✐❝❧❡ ❤❛s ❢♦r ♦❜❥❡❝t✐✈❡ t♦ ♣r❡s❡♥t ❛♥ ❛❧t❡r♥❛✲ t✐✈❡ r❡❝♦♥str✉❝t✐♦♥ ♦❢ ❈♦♥❝❡♣t ♦❢ ❙tr✉❝t✉r❡✱ ♠♦t✐✈❛t❡❞ ❜② t❤❡ ❛rt✐❝❧❡s ❬✶❪ ❛♥❞ ❬✸❪✱ ❛s ❛♥ ❛❜str❛❝t ❣❡♥❡r❛❧✐③❛t✐♦♥ ♦❢ ✇❤❛t ✐s ❛ ♠❛t❤❡♠❛t✐❝❛❧ ♦❜❥❡❝t✳ ❋✐rst✱ ✐ts ❝♦♥str✉❝t✐♦♥ ✐s ♣r❡s❡♥t❡❞✱ t❤❛t ✐s r❡❧❛t❡❞ t♦ t❤❡ ❚②♣❡ ❚❤❡♦r② ❛♥❞ ❖r❞❡r ✐♥ ▲♦❣✐❝✳ ❲❤✐❝❤ ❣✐✈❡ ✉s ❛ ❝♦♥t❡①t ❢♦r ♣r♦♣❡rt✐❡s ❛♥❞ s❡✈❡r❛❧ ✐♥t❡r❡st✐♥❣ ❡①❛♠♣❧❡s✳ ❙❡❝♦♥❞❧②✱ ✇❡ ✇✐❧❧ ♣r♦❝❡❡❞ t♦✇❛r❞s ❛ ❝♦♥✲ ❝r❡t❡ s❡♠❛♥t✐❝ ❢♦r ❛♥❛❧②s✐♥❣ str✉❝t✉r❡s ❛♥❞ ❧❡tt✐♥❣ ✉s ♦♣❡r❛t❡ ✐♥ t❤❡♠✳ ❚❤✉s ✇❡ ❛r❡ ❛❜❧❡ t♦ ❦♥♦✇ ✏✇❤❛t ✐t❳❇ ✐s tr✉❡ ✐♥ ✐t✑✳ ❘❡s✉❧ts ♦❢ ♦r❞❡r ❛♥❞ ✐♥❞✐✈✐❞✉❛❧s r❡❞✉❝t✐♦♥ ❛r❡ ♣r❡s❡♥t❡❞ t❤r♦✉❣❤ ♦✉r ❝♦♥str✉❝t✐♦♥✳ ■♥ t❤❡ ❡♥❞✱ ✇❡ ❢♦r♠❛❧✐③❡❞ t❤❡ ❞✐s❝✉ss✐♦♥s r❡❢❡rr❡❞ ❝♦♠♣❧❡t❡❧② ✐♥ ♦✉r ❚②♣❡ ❚❤❡♦r②✳
❦❡②✇♦r❞s✿ ❙tr✉❝t✉r❡✱ ▼♦❞❡❧s✱ ❙tr✉❝t✉r❡ ❙❡♠❛♥t✐❝s✱ ❍✐❣❤✲❖r❞❡r ▲♦❣✐❝✱ ◆❡✇t♦♥ ❉❛ ❈♦st❛✳
✶❉✐r❡❝❝✐ó♥ ❛❝t✉❛❧✳
✻ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❘❡s✉♠❡♥✿ ❊❧ ❛rt✐❝✉❧♦ t✐❡♥❡ ♣♦r ♦❜❥❡t✐✈♦ ❧❛ r❡❝♦♥str✉❝❝✐ó♥ ❛❧t❡r♥❛t✐✈❛ ❞❡❧ ❝♦♥❝❡♣t♦ ❞❡ ❡str✉❝t✉r❛✱ ♠♦t✐✈❛❞♦ ♣♦r ❧♦s ❛rt✐❝✉❧♦s ❬✶❪ ② ❬✸❪✱ ❝♦♠♦ ✉♥❛ ❣❡♥❡r❛❧✐③❛❝✐ó♥ ❛❜str❛❝t❛ ❞❡ ❧♦ q✉❡ ❡s ✉♥ ♦❜❥❡t♦ ♠❛t❡♠át✐❝♦✳ Pr✐✲ ♠❡r♦✱ ♠♦str❛♠♦s s✉ ❝♦♥str✉❝❝✐ó♥✱ q✉❡ t✐❡♥❡ q✉❡ ✈❡r ❝♦♥ ❧❛ t❡♦rí❛ ❞❡ t✐♣♦s ② ♦r❞❡♥ ❡♥ ❧ó❣✐❝❛✱ ❞❛♥❞♦ ❛ ❧✉❣❛r ❛ ♣r♦♣✐❡❞❛❞❡s ② ✈❛r✐♦s ❡❥❡♠♣❧♦s ✐♥t❡r❡s❛♥t❡s✳ ▲✉❡❣♦ ❛✈❛♥③❛♠♦s ❤❛❝✐❛ ✉♥❛ s❡♠á♥t✐❝❛ ❝♦♥❝r❡t❛✱ ♣❛r❛ s✉ ❛♥á❧✐s✐s✱ ② ♣❛r❛ ♣❡r♠✐t✐r♥♦s ♦♣❡r❛r s♦❜r❡ ❡❧❧❛s✱ s❛❜✐❡♥❞♦ ❞❡ ❡st❡ ♠♦❞♦✱ ❧♦ q✉❡ ❡s ✏❧♦ ✈❡r❞❛❞❡r♦ ❡♥ ❡❧❧❛✑✳ ❖❜t❡♥✐❞♦ ❡❧❧♦✱ ♠♦str❛r❡♠♦s ❧♦s r❡s✉❧t❛✲ ❞♦s ❞❡ r❡❞✉❝❝✐ó♥ ❞❡ ♦r❞❡♥ ② ❞❡ ✐♥❞✐✈✐❞✉♦s✱ ♣❡r♦ ✈✐st♦s ❡♥ ❡st❡ ❝♦♥t❡①t♦✱ ❛sí ❢♦r♠❛❧✐③❛♥❞♦ ❝♦♠♣❧❡t❛♠❡♥t❡ ❡♥ ♥✉❡str❛ t❡♦rí❛ ❞❡ t✐♣♦s ❧❛ ❞✐s❝✉s✐ó♥ ❞❡ ❬✶❪ ✭❱❡r t❛♠❜✐é♥ ❬✷❪ ② ❬✸❪✮ s♦❜r❡ ❡st♦s t❡♠❛s✳
P❛❧❛❜r❛s ❝❧❛✈❡✿ ❊str✉❝t✉r❛s✱ ▼♦❞❡❧♦s✱ ❙❡♠á♥t✐❝❛ ❞❡ ❊str✉❝t✉r❛s✱ ▲ó✲ ❣✐❝❛ ❞❡ ♦r❞❡♥ ♠❛②♦r✱ ◆❡✇t♦♥ ❉❛ ❈♦st❛✳
❮♥❞✐❝❡
■♥tr♦❞✉❝❝✐ó♥ ✼
✶✳ ❘❡❝♦♥str✉❝❝✐ó♥ ❞❡❧ ❝♦♥❝❡♣t♦ ❞❡ ❡str✉❝t✉r❛s ✽
✷✳ ❊❥❡♠♣❧♦s ❞❡ ❡str✉❝t✉r❛s ✷✶
❊str✉❝t✉r❛ s✐♠♣❧❡ ② ♠ú❧t✐♣❧❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ❊str✉❝t✉r❛ ❛♥á❧♦❣❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ❊str✉❝t✉r❛I❢✉♥❞❛♠❡♥t❛❧ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ❊str✉❝t✉r❛ ❝♦♠♣❧❡t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻
✸✳ ▲❡♥❣✉❛❥❡s ② s❡♠á♥t✐❝❛ ♣❛r❛ ❡str✉❝t✉r❛s✳ ✸✼ ▲❡♥❣✉❛❥❡s ❢♦r♠❛❧❡s ♣❛r❛ t✐♣♦s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✼ ❙❡♠á♥t✐❝❛ ♣❛r❛ ❡str✉❝t✉r❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✷ ❚r❛♥s✐❝✐♦♥❡s ❡♥tr❡ ❡str✉❝t✉r❛s✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✻
❘❡❢❡r❡♥❝✐❛s ✻✾
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✼
■♥tr♦❞✉❝❝✐ó♥
❊❧ ♦❜❥❡t✐✈♦ ❞❡❧ ♣r❡s❡♥t❡ tr❛❜❛❥♦ ❡s ❡❧ ❡st✉❞✐♦ ❞❡ ❊str✉❝t✉r❛s ♠♦✲ t✐✈❛❞♦ s✉❜st❛♥❝✐❛❧♠❡♥t❡ ♣♦r ❬✶❪✳ ❊♥ ♣r✐♥❝✐♣✐♦ ❡st❡ ❛rtí❝✉❧♦ ♣✉❡❞❡ s❡r ❧❡í❞♦ ✐♥❞❡♣❡♥❞✐❡♥t❡♠❡♥t❡ ❞❡ ❬✶❪✱ s✐♥ ❡♠❜❛r❣♦ ❧❛ ♠♦t✐✈❛❝✐ó♥ r❡❛❧✱ ②✱ ❡❧ ❡①tr❡♠♦ ❞❡t❛❧❧❡ ❝♦♥ ❡❧ q✉❡ s❡ ❤❛ ❤❡❝❤♦ ❡st❡ ❛rt✐❝✉❧♦✱ s❡ ❛♣r❡❝✐❛ ♠❡❥♦r ♣♦r s✉ ❝♦♥❡①✐ó♥ ❝♦♥ ❡❧ tr❛❜❛❥♦ ❞❡ ◆✳ ❞❛ ❈♦st❛ ② s✉s ❝♦❛✉t♦r❡s✳ ❉❡st❛❝❛✲ ♠♦s q✉❡ ❧❛ ♠♦t✐✈❛❝✐ó♥ ❞❡ ❧♦s ❛✉t♦r❡s ❞❡ ❬✶❪✱ ❬✷❪✱ ❬✸❪✱ ❡s ❡①♣❧✐❝❛r ❞❡s❞❡ ✉♥ ♣✉♥t♦ ❞❡ ✈✐st❛ ❝♦♥t❡♠♣♦rá♥❡♦ ❧❛s ✐♥✈❡st✐❣❛❝✐♦♥❡s ❞❡ ❏✳❙❡❜❛st✐❛♦ ❞❛ ❙✐❧✲ ✈❛ ② ◆✳❦r❛s♥❡r s♦❜r❡ ✉♥❛ t❡♦rí❛ ❞❡ ●❛❧♦✐s ❣❡♥❡r❛❧✐③❛❞❛✳ ▼ás q✉❡ tr❛t❛r ❞❡ ❣❡♥❡r❛❧✐③❛r ❧❛ t❡♦rí❛ ❞❡ ●❛❧♦✐s ❝❧ás✐❝❛ ❬✺❪ ❛ ♦tr♦ t✐♣♦ ❞❡ ❡❝✉❛❝✐♦♥❡s✷♦
♦tr❛s t❡♦rí❛s ❛❧❣❡❜r❛✐❝❛s✸♥♦s♦tr♦s ❡♥t❡♥❞❡♠♦s ❧❛ t❡♦rí❛ ❞❡ ❙✳ ❞❛ ❙✐❧✈❛✹
❝♦♠♦ ✉♥ ✐♥t❡♥t♦ ❞❡ ♣r♦❞✉❝✐r ✉♥❛ t❡♦rí❛ ❛♥á❧♦❣❛ ❡♥ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛✳ P✉❡❞❡ s❡r ✐♠♣♦rt❛♥t❡ ❡♥ ❡st❡ ♣✉♥t♦ ❝✐t❛r ❛ ❆✳❘♦❜✐♥s♦♥✿ ✏t❤❡ s✉❣❣❡st✐♦♥ t❤❛t ♥✉♠❡r♦✉s ✐♠♣♦rt❛♥t ❝♦♥❝❡♣ts ♦❢ ❆❧❣❡❜r❛ ♣♦ss❡ss ♥❛t✉r❛❧ ❣❡r❛❧✐③❛✲ t✐♦♥s t❤❡ ❢r❛♠❡✇♦r❦ ♦❢ t❤❡ ❚❤❡♦r② ♦❢ ▼♦❞❡❧s ❤❛s ♠❡t ✇✐t❤ ❛ r❛t❤❡r ❧❡ss ❧✐✈❡❧② r❡s♣♦♥s❡✳ ◆❡✈❡rt❤❡❧❡ss✱ ✐t ✐s st✐❧❧ t❤❡ ❛✉t❤♦r✬s ❜❡❧✐❡❢ t❤❛t ✐♥✈❡st✐✲ ❣❛t✐♦♥s ✐♥ t❤✐s ❞✐r❡❝t✐♦♥ ❛r❡ ❜♦t❤ ✐♥t❡r❡st✐♥❣ ❛♥❞ ✈❛❧✉❛❜❧❡✳✑❬✽❪ ♣❛❣✐♥❛ ❱■✳
❊♥ ❡st❡ ❛rt✐❝✉❧♦ ♥♦s ♦❝✉♣❛r❡♠♦s ❞❡ ❢♦r♠❛❧✐③❛r ❝♦♠♣❧❡t❛♠❡♥t❡ ❧❛ ♥♦❝✐ó♥ ❞❡ ❡str✉❝t✉r❛ ❛ ❧à ◆✳❞❛ ❈♦st❛ ② s✉s ❝♦❛✉t♦r❡s✱ ② ❞❡❥❛r❡♠♦s ♥✉❡str❛ ✈❡rs✐ó♥ ❞❡ ✏t❡♦r✐❛ ❞❡ ●❛❧♦✐s✑ ♣❛r❛ ♦tr❛ ♣✉❜❧✐❝❛❝✐ó♥✳ ❯♥❛ ♣r✐✲ ♠❡r❛ ✈❡rs✐ó♥ ❞❡ ❡st❛ t❡♦rí❛ ❤❛ ❛♣❛r❡❝✐❞♦ ❡♥ ❬✼❪✳ ❊♠♣❡③❛r❡♠♦s ❝♦♥ ❧❛ ❝♦♥str✉❝❝✐ó♥ ❞❡ ❡str✉❝t✉r❛s ② ❛ ♠❡❞✐❞❛ q✉❡ s❡ ❛✈❛♥❝❡ ❛❜❛r❝❛r❡♠♦s s✉ s❡♠á♥t✐❝❛✳ P❛r❛ ❡❧❧♦ ❡♥ ❡❧ ♣r✐♠❡r ❝❛♣ít✉❧♦ s❡ ♠♦str❛rá ❧❛ ❜❛s❡ ♣❛r❛ ❡❧ ❝♦♥❝❡♣t♦ ❞❡ ❡str✉❝t✉r❛✳ ❈♦♥❡❝tá♥❞♦❧♦ ❝♦♥ ❧♦ ♠♦str❛❞♦ ♣♦r ◆❡✇t♦♥ ❉❛ ❈♦st❛ ❡♥ ❬✶❪✳ ❊♥ ❡❧ s❡❣✉♥❞♦ ❝❛♣ít✉❧♦ ❝♦♥t✐♥✉❛r❡♠♦s ❝♦♥ ❡❥❡♠♣❧♦s ❞❡ ❡str✉❝t✉r❛s✿ ❧❛ ❡str✉❝t✉r❛ s✐♠♣❧❡ ② ♠ú❧t✐♣❧❡✱ ❝♦♠♦ ❧❛ ❡①♣r❡s✐ó♥ ❡♥ ❧❛s ❡str✉❝t✉r❛s ❞❡ ❧❛ ❞✐❢❡r❡♥❝✐❛ ❡ t✐♣♦s ❝♦♥ ✉♥ s♦❧♦ ✐♥❞✐✈✐❞✉♦ ② ✈❛r✐♦s✱ ❧❛ ❡str✉❝t✉r❛ ❛♥á❧♦❣❛ ❝♦♠♦ ❤❡rr❛♠✐❡♥t❛ ❞❡ r❡❞✉❝❝✐ó♥ ❞❡ ✐♥❞✐✈✐❞✉♦s✱ ❧❛ ❡s✲ tr✉❝t✉r❛I❢✉♥❞❛♠❡♥t❛❧ ❝♦♠♦ ❤❡rr❛♠✐❡♥t❛ ❞❡ r❡❞✉❝❝✐ó♥ ❞❡ ♦r❞❡♥✱ ② ♣♦r ✉❧t✐♠♦ ❧❛ ❡str✉❝t✉r❛ ❝♦♠♣❧❡t❛✱ ❝♦♠♦ ❢♦r♠❛❧✐③❛❝✐ó♥ ❞❡❧ ♦❜❥❡t♦ε(D)✱ ❡♥ ❬✶❪ ♣❛❣✐♥❛ ✹✱ q✉❡ ✐♥t❡r❡s❛♥t❡♠❡♥t❡✱ ♥♦ ❡s ❡str✐❝t❛♠❡♥t❡ ✉♥❛ ❡str✉❝t✉r❛
✷❈♦♠♦ ❧♦ ❤✐③♦ ❙✳▲✐❡ ② ❞❡✜♥✐t✐✈❛♠❡♥t❡ ❱❡ss✐♦t ♣❛r❛ ❡❝✉❛❝✐♦♥❡s ❞✐❢❡r❡♥❝✐❛✲ ❧❡s✳
✸❈♦♠♦ ❡♥ ❡❧ ❝❛s♦ ❞❡ ❑r❛s♥❡r ✹❈♦♠♦ ❡①♣❧✐❝❛❞❛ ❡♥ ❬✶❪✳
✽ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❡♥ ❡❧ s❡♥t✐❞♦ ❞❛❞♦ ❛q✉í✳ ❊♥ ❡❧ t❡r❝❡r ❝❛♣ít✉❧♦ ❝♦♥str✉✐♠♦s ✉♥❛ s❡♠á♥✲ t✐❝❛ ♣❡rt✐♥❡♥t❡ ♣❛r❛ ❡❧❧♦ r❡❝♦♥str✉✐♠♦s ❧♦s ❧❡♥❣✉❛❥❡s✱ ❝♦♥ ❧❛ s❛❧✈❡❞❛❞ ❞❡ q✉❡ ❛❤♦r❛ s♦♥ ♣❛r❛ ✈❛r✐♦s ✐♥❞✐✈✐❞✉♦s✳ ❆ ♣❛rt✐r ❞❡ ❡❧❧♦ ❢♦r♠❛❧✐③❛r❡♠♦s ❧❛s ❤❡rr❛♠✐❡♥t❛s ♣❛r❛ ❧❛ ✐♥t❡r♣r❡t❛❝✐ó♥✱ ② ❛sí ❛✈❛♥③❛r ❤❛❝✐❛ ✉♥❛ t❡♦rí❛ ❞❡ ♠♦❞❡❧♦s ♣❛r❛ ❡str✉❝t✉r❛s✳ ❍❛❜✐❡♥❞♦ ❞❡s❛rr♦❧❧❛❞♦ ❡st❛s ❤❡rr❛♠✐❡♥t❛s ❡s❡♥❝✐❛❧❡s✱ ♠♦str❛r❡♠♦s ✉♥ ❝♦♥❝❡♣t♦ ❜❛st❛♥t❡ ✐♥t✉✐t✐✈♦✺✱ ❛❧ q✉❡ ❧❧❛♠❛✲
r❡♠♦s tr❛♥s✐❝✐ó♥✳ ❙✉♠❛❞♦ ❛ ❧❛s ❤❡rr❛♠✐❡♥t❛s ♠❡♥❝✐♦♥❛❞❛s✱ ♣♦s✐❜✐❧✐t❛rá ✉♥❛ ❞❡♠♦str❛❝✐ó♥ ❞❡ ♥✉❡str♦s ♣r✐♥❝✐♣❛❧❡s t❡♦r❡♠❛s✱ r❡❞✉❝❝✐ó♥ ❞❡ ✐♥❞✐✲ ✈✐❞✉♦s ✭✸✳✶✾✮✱ r❡❞✉❝❝✐ó♥ ❞❡ ♦r❞❡♥ ✭✸✳✷✷✮ ② ❝♦r♦❧❛r✐♦ s♦❜r❡ r❡❞✉❝❝✐ó♥ ❞❡ ♦r❞❡♥ ❡ ✐♥❞✐✈✐❞✉♦s ✭✸✳✷✸✮✳
✶✳ ❘❡❝♦♥str✉❝❝✐ó♥ ❞❡❧ ❝♦♥❝❡♣t♦ ❞❡ ❡str✉❝✲
t✉r❛s
▲♦ ❛ r❡❛❧✐③❛r ❡♥ ❡st❡ tr❛❜❛❥♦ s❡r❛ ❜❛❥♦ ❧❛ t❡♦rí❛ ❞❡ ❝♦♥❥✉♥t♦s ❞❡ ◆❡✉♠❛♥✲❇❡r♥❛②s✲●ö❞❡❧✭◆❇●✮✱ ❛✉♥q✉❡ ❧❛ ❡①♣♦s✐❝✐ó♥ s❡❛ ✉♥ t❛♥t♦ ✐♥✲ ❢♦r♠❛❧✱ ❡st❛ s❡r❛ r✐❣✉r♦s❛ ② ♣r❡❝✐s❛✱ ❡♥ ❡❧ s❡♥t✐❞♦ ✉s✉❛❧ ❞❡ ❧♦s tr❛❜❛❥♦s ❞❡ ❧ó❣✐❝❛ ♠❛t❡♠át✐❝❛✿ ✉s❛♠♦s ❛r❣✉♠❡♥t♦s ❞❡ ♠❛t❡♠át✐❝❛ ❡stá♥❞❛r ❡♥ ❡❧ ❡♥t❡♥❞✐♠✐❡♥t♦ q✉❡ t♦❞♦ s❡ ♣✉❡❞❡ ❢♦r♠❛❧✐③❛r ❡♥ ❡❧ ❧❡♥❣✉❛❥❡ ❛❞❡❝✉❛❞♦✳ ❉❡✜♥✐❝✐ó♥ ✶✳✶✳ ❙❡❛ ✉♥ ❝♦♥❥✉♥t♦ I6=∅❞❡✜♥✐♠♦s✿
✶✳ ❊❧ ❣❡♥❡r❛❞♦ ❞❡ I✱ ❝♦♠♦ ❡❧ ❝♦♥❥✉♥t♦
G(I) ={ha1, . . . , ani|a1, . . . , an∈I∧1≤n < ω},
❞♦♥❞❡ ω ❡s ❡❧ ♠❡♥♦r ♦r❞✐♥❛❧ tr❛♥s✜♥✐t♦✳ ❆❞❡♠ás ❞❡✜♥✐♠♦s ❞❡ ♠❛♥❡r❛ r❡❝✉rs✐✈❛✱ ♣❛r❛ 0≤l < ω✱
G(0)(I) =I, ②✱G(l)(I) =G[{ G(j)(I)|0≤j < l}.
✷✳ ❊❧ ❝♦♥❥✉♥t♦ TI✱ ❞❡ ❧❛ ❢♦r♠❛
TI =
[ n
G(l)(I)|0≤l < ωo,
❉✐r❡♠♦s q✉❡ ❡❧ ❝♦♥❥✉♥t♦ TI✱ q✉❡ s❛t✐s❢❛❝❡ I∩ G(l)(I) = ∅✱ ♣❛r❛
❝✉❛❧q✉✐❡r l∈N✱ ❡s ❡❧ ❝♦♥❥✉♥t♦ ❞❡ t✐♣♦s ❞❡ I✳
✺❈❧❛r❛♠❡♥t❡ ✈✐st♦ ❡♥ ♦tr♦s á♠❜✐t♦s✱ ♣♦r ❡❥❡♠♣❧♦ t❡♦rí❛ ❞❡ ♠♦❞❡❧♦s s❛t✉✲ r❛❞❛✱ ♣❡r♦ ❞❡ ✉♥ ♠♦❞♦ ❞✐st✐♥t♦✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✾
✸✳ ▲♦s ❝♦♥❥✉♥t♦sT(l)(I)✱ ❝♦♥ 0≤l < ω✱
T(0)(I) =G(0)(I), ②✱T(l)(I) =G(l)(I)\G(l−1)(I).
❉✐r❡♠♦s q✉❡ T(l)(I) ❡s ❡❧ ❝♦♥❥✉♥t♦ ❞❡ t✐♣♦s ❞❡ ♦r❞❡♥ l ❞❡
TI✱ ② ❧❧❛♠❛r❡♠♦s ❛T(0)(I)❡❧ ❝♦♥❥✉♥t♦ ❞❡ ✐♥❞✐✈✐❞✉♦s ❞❡TI✳
◆♦t❡♠♦s q✉❡ ❡st❛ ❞❡✜♥✐❝✐ó♥ ❡s ❡q✉✐✈❛❧❡♥t❡ ❛ ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡ ❧♦s t✐♣♦s T ❡♥ ❬✶❪ ♣á❣✐♥❛ ✹✳ P❛r❛ ❡❧❧♦ ♦❜s❡r✈❡♠♦s q✉❡ ❡♥ ❡❧ ❝❛s♦ I ={i}✱ a∈T ❞❡ ♦r❞❡♥✻0❡♥t♦♥❝❡sa∈I⊂TI✱ ❧✉❡❣♦ ✉s❛♥❞♦ ❧❛ ❢♦r♠❛ ✐♥❞✉❝t✐✈❛
❞❡Tt❡♥❡♠♦s q✉❡a=ha1, . . . , ani ∈T✱ ② ✉s❛♥❞♦ ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥
s♦❜r❡ ❡❧ ♦r❞❡♥ ❞❡ai♣❛r❛1≤i≤m✱ t❡♥❡♠♦sai∈TI✳ ▲✉❡❣♦ai∈ Gni(I)
♣❛r❛ ❝✐❡rt♦s 0 ≤ ni < ω✳ ❆sí✱ ♣♦r ❧❛ ❞❡✜♥✐❝✐ó♥ ❛♥t❡r✐♦r✱ ❜❛st❛ ✜❥❛r q
❝♦♠♦ ❡❧ ♠á①✐♠♦ ❞❡ ❧♦sni✱ ♣❛r❛ ♦❜t❡♥❡r q✉❡
ha1, . . . , ani ∈ G(q+1)(I)⊂TI.
❆sí T ⊂TI✳ P♦r ♦tr♦ ♦tr♦ ❧❛❞♦ ❡s ✉♥ t❛♥t♦ ❢á❝✐❧ ✈❡r q✉❡ TI ⊂T✱ ♣❛r❛
❡❧❧♦ ✈❡r q✉❡ G(0)(I)⊂T ② q✉❡ s✐G(q)(I)⊂T ♣❛r❛ ❝✉❛❧q✉✐❡rq < l < ω
❡♥t♦♥❝❡s G(l)(I)⊂T✳ ❆sí✱ ❝♦♠♦l ❡s ❛r❜✐tr❛r✐♦✱ t❡♥❡♠♦sT
I ⊂T✳
Pr♦♣♦s✐❝✐ó♥ ✶✳✷✳ ❙❡❛ 1≤l < ω✳ ❚❡♥❡♠♦s✿ ✶✳ G(l)(I) ❡s ✉♥ ❝♦♥❥✉♥t♦✳
✷✳ ▲❛ ❝❧❛s❡{G(j)(I)|0≤j < ω} ❡s ✉♥ ❝♦♥❥✉♥t♦✳
✸✳ P❛r❛1≤l≤l′ < ω✱G(l)(I)⊂ G(l′
)(I)✳
✹✳ ◆♦ s✐❡♠♣r❡ s❡ ❝✉♠♣❧❡ q✉❡ I∩ G(1)(I) = ∅✱ ② ♣♦r ❧♦ t❛♥t♦ ❧❛
r❡str✐❝❝✐ó♥ ❤❡❝❤❛ ❡♥ ❧❛ ♣❛rt❡ ✷ ❞❡ ❞❡✜♥✐❝✐ó♥ ✶✳✶ t✐❡♥❡ s❡♥t✐❞♦ ♣❛r❛ ✉♥ ❝♦♥❥✉♥t♦ ❛r❜✐tr❛r✐♦✳
✺✳ T(l)(I) ❡s ✉♥ ❝♦♥❥✉♥t♦✳
✻✳ ▲♦s ❝♦♥❥✉♥t♦sG(l)(I)✱ ②✱T(l)(I)✱ s♦♥ ❞✐st✐♥t♦s ❞❡❧ ✈❛❝í♦✳
✼✳ ❙❡❛♥ 0≤i≤j < ω✱ s❡ ❝✉♠♣❧❡✿ T(i)(I)∩T(j)(I) = ∅✱ s✐ ② s♦❧♦
s✐✱ i6=j✳
✽✳ TI ❡s ✉♥ ❝♦♥❥✉♥t♦ ② TI =S{T(j)(I)|1≤j < ω}✳
✻❊♥ ❡❧ s❡♥t✐❞♦ q✉❡ s❡ ❧❡ ❞❛ ❛❧ ✏♦r❞❡♥✑ ❡♥ ❡❧ tr❛❜❛❥♦ ❬✶❪ ♣❛❣✐♥❛ ✹✳
✶✵ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❉❡♠♦str❛❝✐ó♥✳ ❊♥✉♠❡r❛♥❞♦ ❧❛s ❞❡♠♦str❛❝✐♦♥❡s✿
✶✳ ❙✐ I ❡s ✉♥ ❝♦♥❥✉♥t♦ ❡♥t♦♥❝❡s G(1)(I) ❝♦♥s✐❞❡r❛❞♦ ❝♦♠♦ ❧❛s ❢✉♥✲
❝✐♦♥❡s ❞❡s❞❡ ✉♥ s✉❜❝♦♥❥✉♥t♦ ❞❡ N ❛ I✱ ❡s ❞❡❝✐r✱ x ∈ G(1)(I)
❡♥t♦♥❝❡s x∈P(N×I)✱ ❞♦♥❞❡P ②×s♦♥ sí♠❜♦❧♦s ♣❛r❛ ❡❧ ❝♦♥✲
❥✉♥t♦ ♣♦t❡♥❝✐❛ ② ❡❧ ♣r♦❞✉❝t♦ ❝❛rt❡s✐❛♥♦✱ r❡s♣❡❝t✐✈❛♠❡♥t❡✳ ❆sí
G(1)(I)⊂P(N×I)✱ ❞❡ ❧♦ q✉❡ s❡ ❞❡❞✉❝❡ q✉❡G(1)(I)❡s ✉♥ ❝♦♥✲
❥✉♥t♦✼✳ P♦r ♦tr♦ ❧❛❞♦ ✈❡❛♠♦s q✉❡ ♣❛r❛1≤l < ω✱ t❡♥❡♠♦s q✉❡✿
x∈ G(l)(I)✱ ❡♥t♦♥❝❡sx⊂Nl× G(l−1)(I)✳ ❯s❛♥❞♦ ✉♥❛ ♠❛q✉✐♥❛❞❛
❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥ t❡♥❡♠♦sG(l−1)(I)❡s ✉♥ ❝♦♥❥✉♥t♦✱ ❛sí✱x∈
P(Nl×G(l−1)(I))✱ ❞❡ ❧♦ q✉❡ s❡ ❞❡❞✉❝❡G(l)(I)⊂P(Nl×G(l−1)(I))✳
❉❡ ❧❛ ♠✐s♠❛ ❢♦r♠❛ ❡♥ q✉❡ s❡ ❞❡❞✉❝❡ q✉❡G(1)(I)❡s ✉♥ ❝♦♥❥✉♥t♦✱
❞❡❞✉❝✐♠♦sG(l)(I)✱ ♣❛r❛ 1≤l < ω✳
✷✳ ◆♦t❡♠♦s q✉❡ ❧❛ ❝❧❛s❡{G(j)(I)|0≤j < ω}❡s ✉♥ ❝♦♥❥✉♥t♦✱ ♣✉❡st♦
q✉❡ ❝❧❛r❛♠❡♥t❡ ❤❛② ✉♥❛ ❛♣❧✐❝❛❝✐ó♥ s♦❜r❡②❡❝t✐✈❛ ❞❡N0❛ t❛❧✽✳
✸✳ ❙✐ A ⊂B ❡♥t♦♥❝❡s ❝❧❛r❛♠❡♥t❡✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ❛♥t❡r✐♦r✱ G(A)⊂ G(B)✳ ❆❞❡♠ás ✈❡❛♠♦s q✉❡
[
{G(j)(I)|0≤j < l} ⊂[
{G(j)(I)|0≤j < l′
},
♣❛r❛l, l′ ❞❡✜♥✐❞♦s ♣❛r❛ ❡st❡ ít❡♠✳ ❆❤♦r❛ ✉s❛♥❞♦ ❧♦ ♠♦str❛❞♦ ❡♥ ✉♥ ♣r✐♥❝✐♣✐♦
G([{G(j)(I)|0≤
j < l})⊂ G([{G(j)(I)|0≤
j < l′}),
❝❧❛r❛♠❡♥t❡ t❡♥❡♠♦s q✉❡G(l)(I)⊂ G(l′
)(I)✱ ♣❛r❛1≤l≤l′✳
✹✳ ❊s❝♦❥❛♠♦s I = {i1, i2, fI}✱ ❝♦♥ fI : {1,2} 7→ {i1, i2}✱ ❞❡ ♠♦❞♦
q✉❡fI(j) =ij✱ ❡♥ t❛❧ ❝❛s♦✱ fI =hi1, i2i✳ ▲✉❡❣♦ ❝❧❛r❛♠❡♥t❡fI ∈
G(1)(I)✱ ❛sí✱I∩ G(1)(I)6=∅✳
✺✳ ❈❧❛r❛♠❡♥t❡T(l)(I)⊂ G(l)(I)✱ ❧✉❡❣♦ ♣♦r ❡❧ ít❡♠ ✶ ❞❡ ❡st❛ ♣r♦♣♦✲
s✐❝✐ó♥T(l)(I)❡s ✉♥ ❝♦♥❥✉♥t♦✾✳
✼P❛r❛ ♠❛s ❞❡t❛❧❧❡ ✈❡r ♣á❣✐♥❛ ✷✸✹✱ t❡♦r❡♠❛ ✽✳✽ ❞❡❧ ❧✐❜r♦ ❬✹❪✳ ✽P❛r❛ ♠❛s ❞❡t❛❧❧❡ ✈❡r ♣á❣✐♥❛ ✷✸✹✱ t❡♦r❡♠❛ ✽✳✼ ❞❡❧ ❧✐❜r♦ ❬✹❪✳ ✾P❛r❛ ♠❛s ❞❡t❛❧❧❡ ✈❡r ♣á❣✐♥❛ ✷✸✹✱ t❡♦r❡♠❛ ✽✳✽ ❞❡❧ ❧✐❜r♦ ❬✹❪✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✶✶
✻✳ ◆♦t❡♠♦s q✉❡ G(0)(I) =I 6=∅✳ P♦r ♦tr♦ ❧❛❞♦✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶
ít❡♠ ✶✱
G(l+1)(I) =G([
{G(j)(I)|1≤j < l+ 1}),
♣♦r ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥ t❡♥❡♠♦s q✉❡ G(l)(I) 6= ∅✱ ❛sí✱ ♣♦r
❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✶ t❡♥❡♠♦s q✉❡ ❡①✐st❡a∈ G(l)(I)✱ ❞❡ ♠♦❞♦ q✉❡
hai ∈ G(l+1)(I)✳ ❆síG(l+1)(I) 6=∅✳ P❛r❛ ❞❡♠♦str❛r q✉❡T(l)(I)
♥♦ ❡s ✈❛❝í♦ ♥♦t❡♠♦s ❛❧❣✉♥♦s ❤❡❝❤♦s✿
❛✮ ❙✐ x6∈I✱ ❡♥t♦♥❝❡shxi 6∈ G(1)(I)✳ ❊st❡ ❤❡❝❤♦ ❡st❛ ❞❛❞♦ ♣♦r
❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✶✳
❜✮ ❉❡❧ ♠✐s♠♦ ♠♦❞♦✱ ❞❡❞✉❝✐♠♦s✱ ♣♦r ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✷✱ x6∈S{G(j)(I)|0≤j < l}✱ ❡♥t♦♥❝❡s hxi 6∈ G(l)(I)✳
❝✮ ◆♦t❡♠♦s q✉❡ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ I ∩ G(1)(I) = ∅✱ ②❛ q✉❡
G(1)(I)6=∅✱ ❡♥t♦♥❝❡s ❡①✐st❡x∈ G(1)(I)✱ ❞❡ ♠♦❞♦ q✉❡x6∈I✱
❧✉❡❣♦T(1)(I)6=∅✳
❯s❛♥❞♦ ❝♦♠♦ ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥T(l)(I)6=∅✱ ♣❛r❛1≤l < ω✱
♦❜t❡♥❡♠♦s q✉❡✿ ❡①✐st❡ x∈ G(l)(I)✱ ❞❡ ♠♦❞♦ q✉❡ x6∈ G(l−1)(I)✳
❆❤♦r❛✱ ❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✷ ② ❡❧ ít❡♠ ✸ ❞❡ ❡st❛ ♣r♦♣♦s✐❝✐ó♥✱ x6∈S{G(j)(I)|0 ≤j < l}✳ ▲✉❡❣♦ ♣♦r ❡❧ ít❡♠b)✱ t❡♥❡♠♦shxi 6∈
G(l)(I)✱ ♣❡r♦ ❛ s✉ ✈❡③✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ hxi ∈ G(l+1)(I)✳ P♦r ❧♦
t❛♥t♦ T(l+1)(I) =G(l+1)(I)\G(l)(I)6=∅✳
❊♥ sí♥t❡s✐sT(l)(I)6=∅✳ P♦r ❝✐❡rt♦✱T(0)(I)❡s ❝❧❛r❛♠❡♥t❡ ❞✐st✐♥t♦ ❞❡❧ ✈❛❝í♦ ❞❛❞❛ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✶✳
✼✳ ❉❡ ❞❡r❡❝❤❛ ❛ ✐③q✉✐❡r❞❛✳ ❙✐ i 6= j✱ ❡♥t♦♥❝❡s i < j✱ ❧✉❡❣♦✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✷✱
T(i)(I)⊂ G(i)(I)⊂ G(j−1)(I),
❛sí ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✸T(j)(I)∩T(i)(I) =∅✳
❉❡ ✐③q✉✐❡r❞❛ ❛ ❞❡r❡❝❤❛✱ s✉♣♦♥❣❛♠♦s q✉❡ i = j✱ ②✱ T(j)(I)∩
T(i)(I) = ∅✱ ❧✉❡❣♦ t❡♥❡♠♦s ✉♥ ❝❧❛r♦ ❝♦♥tr❛s❡♥t✐❞♦ ❝♦♥ ❡❧ ít❡♠
❛♥t❡r✐♦r ② ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✶✳
✽✳ P♦r ❡❧ ít❡♠ ✷ ❞❡ ❡st❛ ♣r♦♣♦s✐❝✐ó♥TI ❡s ✉♥ ❝♦♥❥✉♥t♦✳ ❱❡❛♠♦s q✉❡
T(0)(I) =G(0)(I)✳ ❆❤♦r❛ s✉♣♦♥❞r❡♠♦s q✉❡ ♣❛r❛1≤l < ω✱
[
{G(j)(I)|0≤j < l} ⊂[{T(j)(I)|0≤j < l},
✶✷ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❧✉❡❣♦
[
{G(j)(I)|0≤
j < l} ∪ G(l)(I) =[
{T(j)(I)|0≤j < l} ∪ G(l)(I),
❈♦♠♦
G(l−1)(I)⊂[
{G(j)(I)|0≤j < l}✱ ②✱T(l)(I)∪G(l−1)(I) =G(l)(I),
❞❡❞✉❝✐♠♦s
[
{T(j)(I)|0≤j < l} ∪T(l)(I) =[{T(j)(I)|0≤j < l+ 1},
P♦r ✉❧t✐♠♦✱ ❝♦♠♦
[
{G(j)(I)|0≤
j < l+ 1}=[{G(j)(I)|0≤
j < l} ∪ G(l)(I),
② q✉❡l∈N0 ❡s ❛r❜✐tr❛r✐♦✱TI =S{T(j)(I)|0≤j < ω}✳
❉❡✜♥✐❝✐ó♥ ✶✳✸✳ ❙❡❛♥l❞❡ ♠♦❞♦ q✉❡0≤l < ω✱ ② s❡❛a∈TI✳ ❉❡✜♥✐♠♦s
❧❛ ❢✉♥❝✐ó♥ ordI :TI 7→N0✱ ❞❡ ❧❛ ❢♦r♠❛✱
ordI(a) =l✱ s✐ ② s♦❧♦ s✐✱ a∈T(l)(I).
▲❧❛♠❛r❡♠♦s ❛ ordI(a) ❡❧ ♦r❞❡♥ ❞❡a✳
◆♦t❡♠♦s q✉❡ ❡❧ ♦r❞❡♥ ❞❡ a❡st❛ ❜✐❡♥ ❞❡✜♥✐❞♦ ❞❛❞❛ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ❛♥t❡r✐♦r✱ ❡♥ ❡s♣❡❝✐✜❝♦ ❡❧ ít❡♠ ✼✳
▲❛ s✐❣✉✐❡♥t❡ ♣r♦♣♦s✐❝✐ó♥ ♠✉❡str❛ q✉❡ ♥✉❡str❛ ❞❡✜♥✐❝✐ó♥ ❡s ✉♥❛ ❣❡✲ ♥❡r❛❧✐③❛❝✐ó♥ ♥❛t✉r❛❧ ❞❡❧ ✏♦r❞❡♥✑ ❞❡ ❬✶❪ ♣á❣✐♥❛ ✹✳
Pr♦♣♦s✐❝✐ó♥ ✶✳✹✳ ❙❡❛♥ a, a1, . . . , an ∈ TI✱ ② s❡❛ n ❞❡ ♠♦❞♦ q✉❡ 1 ≤
n < ω✳ ❙❡ ❝✉♠♣❧❡
✶✳ a∈I✱ ❡♥t♦♥❝❡s ordI(a) = 0✳
✷✳ a=ha1, . . . , ani✱ ❡♥t♦♥❝❡s
ordI(a) = m´ax{ordI(a1), . . . , ordI(an)}+ 1.
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✶✸
❉❡♠♦str❛❝✐ó♥✳ ❙✐ a∈I✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✸✱a∈T(0)(I)✱ ②✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✸✱ordI(a) = 0✳ P❛r❛a∈T(1)(I)✱ t❡♥❞rí❛♠♦s✱ ♣♦r ❞❡✜♥✐❝✐ó♥
✶✳✶ ít❡♠ ✸✱{a1, . . . , an} ⊂T(0)(I)✱ ❛sí ❝❧❛r❛♠❡♥t❡ s❡ ❝✉♠♣❧❡
ordI(a) = 0 + 1 = m´ax{ordI(a1), . . . , ordI(an)}+ 1.
P❛r❛a∈T(j)(I)✱ ❝♦♥1< j < ω✱ t❡♥❡♠♦s ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✶ ít❡♠ ✷✱
{a1, . . . , an} ⊂
[
{G(i)(I)|0≤i < j},
❧✉❡❣♦ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✸✿ ordI(ai) ≤ j −1✱ ♣❛r❛ 1 ≤ i ≤ n✳ ❉❛❞❛ ❧❛
❢♦r♠❛ ❞❡as❛❜❡♠♦sha1, . . . , ani 6∈ G(j−1)(I)✱ ❛sí
{a1, . . . , an} 6⊂
[
{G(i)(I)|0≤i < j−1},
❡♥ sí♥t❡s✐s✿ ❡①✐st❡ aq✱ ❝♦♥1≤q≤n✱ q✉❡ ❝✉♠♣❧❡✱
aq 6∈
[
{G(i)(I)|0≤
i < j−1}✱ ②✱aq ∈
[
{G(i)(I)|0≤
i < j}.
▲✉❡❣♦ aq ∈ G(j−1)(I)✱ ②✱aq 6∈ G(j−2)(I)✱ ❛sí✱ aq ∈ T(j−1)(I)✱ ❡s ❞❡❝✐r
ordI(aq) = j −1✳ ❈♦♠♦ ordI(ai) ≤ j −1✱ ♣❛r❛ 1 ≤ i ≤ n✱ ❡♥t♦♥✲
❝❡s✱ ordI(aq) = m´ax{ordI(a1), . . . , ordI(an)}✳ P♦r ❧♦ t❛♥t♦✱ ❞❛❞♦ q✉❡
ordI(a) =j✱
ordI(a) =j =ordI(aq) + 1 = m´ax{ordI(a1), . . . , ordI(an)}+ 1.
❆❤♦r❛ q✉❡ ❞✐s♣♦♥❡♠♦s ❞❡ ✉♥❛ ❜✉❡♥❛ ❞❡✜♥✐❝✐ó♥ ❞❡ t✐♣♦s ♣♦❞❡♠♦s ❞❡✜♥✐r t✐♣♦s ❞❡ ❝♦♥❥✉♥t♦s✳ ▲✉❡❣♦ ✉s❛r❡♠♦s ❡st♦ ♣❛r❛ ❞❡✜♥✐r t✐♣♦s ❞❡ ❡str✉❝t✉r❛s✳
❉❡✜♥✐❝✐ó♥ ✶✳✺✳ ❙❡❛Obj ✉♥ ❝♦♥❥✉♥t♦✳
✶✳ ❉✐r❡♠♦s q✉❡Obj ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡ I s✐ ❝✉♠♣❧❡✿ ❛✮ ❊①✐st❡ f ❜✐②❡❝❝✐ó♥ ❡♥tr❡ Obj ❡I✱f :I7→Obj✳
❜✮ x∈Obj✱ ❡♥t♦♥❝❡s x6=∅✳
❝✮ ♣❛r❛x, y∈Obj✱x6=y s✐ ② s♦❧♦ s✐ x∩y=∅✳
✶✹ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
✷✳ P❛r❛✿ Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡ I✱ f ✉♥❛ ❜✐②❡❝❝✐ó♥ ❝♦rr❡s✲ ♣♦♥❞✐❡♥t❡ ❡♥tr❡ I ② Obj✱ ❞❡✜♥✐♠♦s OI✱ ❧❛ ♦❜❥❡t✐✈❛❝✐ó♥ ❞❡ I✱
❞❡❧ s✐❣✉✐❡♥t❡ ♠♦❞♦
❛✮ ♣❛r❛ a∈I✱OI(Obj)(a) =f(a)× {0}✳
❜✮ s✐a=ha1, . . . , ani✱ ❝♦♥a1, . . . , an∈TI✱ ❡♥t♦♥❝❡s
OI(Obj)(a) = (P(OI(Obj)(a1)×. . .×OI(Obj)(a1))\{∅})×{ordI(a)}.
❝✮ ❙❡❛ l∈N0✱ ❞❡✜♥✐♠♦sP(l)
I ✱ ❡❧ ♣❡❧❞❛ñ♦ l ❞❡ I✱ ❝♦♠♦
PI(l)(Obj) =[{OI(Obj)(a)|a∈T(l)(I)}.
✸✳ ❉❡✜♥✐♠♦sEI ❧❛ ❡s❝❛❧❡r❛ ❞❡ I✱ ❞❡ ❧❛ ❢♦r♠❛
EI(Obj) =
[
{OI(Obj)(a)|a∈TI}.
✹✳ ❉❡✜♥✐♠♦s∅ω✱ ❧❛ r❡❧❛❝✐ó♥ ✈❛❝í❛✱ ❞❡ ❧❛ ❢♦r♠❛ ∅ω= (∅, ω)✳
P♦r ❝✐❡rt♦ ❡♥ ❡❧ ♠❛r❝♦ ❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ❛♥t❡r✐♦r ít❡♠ ✶✱ s✐Obj❝✉♠♣❧❡ 1.a) ② 1.b) ♥♦ ♥❡❝❡s❛r✐❛♠❡♥t❡ ❝✉♠♣❧❡ ❧❛ ♣r♦♣✐❡❞❛❞ 1.c)✱ ❡♥ t❛❧ ❝❛s♦ ♣♦❞❡♠♦s ✉s❛rObj∗={x∗|x∈Obj}✱ ❞♦♥❞❡x∗=x× {x}✳ ◆♦t❡♠♦s q✉❡
♣❛r❛ ❝❛❞❛a∗, b∗ ❞✐st✐♥t♦s ❡♥tr❡ sí s❡ ❝✉♠♣❧❡a∗∩b∗=∅✳
◆♦t❡♠♦s q✉❡PI(l)(Obj)❡s ✉♥ ❝♦♥❥✉♥t♦✱ ♣✉❡st♦ q✉❡{OI(Obj)(a)|a∈
T(l)(I)} ❡s ✉♥ ❝♦♥❥✉♥t♦✳ ❊st♦ s❡ ❞❡❞✉❝❡ ❞❡ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✷ ít❡♠ ✺
② ❡❧ ❤❡❝❤♦ ❞❡ q✉❡ ❤❛② ✉♥❛ ❛♣❧✐❝❛❝✐ó♥ ❞❡s❞❡T(l)(I)❛{O
I(Obj)(a)|a∈
T(l)(I)} q✉❡ ❡s s♦❜r❡②❡❝t✐✈❛✶✵✳ ❉❡❧ ♠✐s♠♦ ♠♦❞♦ ❞❡❞✉❝✐♠♦s q✉❡
{PI(l)(Obj)|0≤l < ω},②✱EI(Obj)✱ s♦♥ ❝♦♥❥✉♥t♦s✳
❈♦♥ r❡s♣❡❝t♦ ❛❧ ít❡♠ ✸ ❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ❛♥t❡r✐♦r ♥♦t❡♠♦s q✉❡EI(Obj)
❡s ❡q✉✐✈❛❧❡♥t❡ t❛♠❜✐é♥
EI(Obj) =
[
{PI(l)|l∈ω},
② ❡❧ ♠♦t✐✈♦ ❞❡ ❞❡✜♥✐r❧♦✱ ❛❧ ♠♦❞♦ q✉❡ ❢✉❡✱ ❡s ♠❛s q✉❡ ♥❛❞❛ ❞❡❥❛r ❡♥ ❝❧❛r♦ q✉❡ EI(Obj)❡s ❡❧ ❝♦♥❥✉♥t♦ ❞❡ t♦❞❛s ❧❛s r❡❧❛❝✐♦♥❡s ♣♦s✐❜❧❡s✳
◆♦ ❡s ❞✐❢í❝✐❧ ♥♦t❛r q✉❡ ✉♥ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ❧♦ ❞❡✜♥✐❞♦ ❡sI={i}
②Obj ={D} ♣❛r❛ ✉♥D ❝♦♥❥✉♥t♦ ❝✉❛❧q✉✐❡r❛✱ ❡st❡ ❝❛s♦ ❡s ✐♠♣♦rt❛♥t❡✱
✶✵P❛r❛ ♠❛s ❞❡t❛❧❧❡ ✈❡r ♣á❣✐♥❛ ✷✸✹✱ t❡♦r❡♠❛ ✽✳✼ ❞❡❧ ❧✐❜r♦ ❬✹❪✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✶✺
♣✉❡st♦ q✉❡ ♥♦s ♣❡r♠✐t✐rá ❝♦♥❡❝t❛r ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡ ❡str✉❝t✉r❛ q✉❡ s❡ ❞❛rá ♠❛s ❛❞❡❧❛♥t❡ ❝♦♥ ❡❧ ❝♦♥❝❡♣t♦ ❞❡ ❡str✉❝t✉r❛ ❡♥ ◆❡✇t♦♥ ❞❛ ❈♦st❛✶✶✳
Pr♦♣♦s✐❝✐ó♥ ✶✳✻✳ ❙❡❛♥ Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡ I✱ a, b ∈ TI✱
i, j∈N0 ②∅ω ❧❛ r❡❧❛❝✐ó♥ ✈❛❝í❛✳ ❙❡ ❝✉♠♣❧❡
✶✳ OI(Obj)(a)6=∅
✷✳ ∅ω6∈ O
I(Obj)(a)✳
✸✳ a6=b s✐ ② s♦❧♦ s✐OI(Obj)(a)∩ OI(Obj)(b) =∅✳
✹✳ a6=b ❡♥t♦♥❝❡sOI(Obj)(a)6=OI(Obj)(b)✳✶✷
✺✳ j6=i✱ s✐ ② s♦❧♦ s✐✱ P(j)(Obj)∩ P(i)(Obj) =∅✳
❉❡♠♦str❛❝✐ó♥✳ ❊♥✉♠❡r❛♥❞♦ ❧❛s ❞❡♠♦str❛❝✐♦♥❡s✿
✶✳ ❉❡♠♦str❛r❡♠♦s ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ ♦r❞❡♥ ❞❡a✳ ❙✐ordI(a) = 0✱
❡♥t♦♥❝❡sa∈I✳ ▲✉❡❣♦ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✶ ② ✷
OI(Obj)(a) =f(a)× {0} 6=∅.
❆❤♦r❛✱ ♣❛r❛ ordI(a) =m✱ t❡♥❡♠♦s
OI(Obj)(a) = (P(OI(Obj)(a1)×. . .×OI(Obj)(an))\{∅})×{ordI(a)},
❞♦♥❞❡a=ha1, . . . , ani✳ ◆♦t❡♠♦s q✉❡ ♣♦r ♣r♦♣♦s✐❝✐ó♥ ✶✳✹ ít❡♠ ✷
ordI(ai)< m✱ ♣❛r❛ 1 ≤ i≤ n✱ ❛sí✱ ♣♦r ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥✱
t❡♥❡♠♦s q✉❡OI(Obj)(ai)6=∅✳ ▲✉❡❣♦
P(OI(Obj)(a1)×. . .× OI(Obj)(an))\{∅} 6=∅.
P♦r ❧♦ t❛♥t♦✱ ❡♥ ✈✐rt✉❞ ❞❡ ❧❛ ✐❣✉❛❧❞❛❞❡s ♠♦str❛❞❛sOI(Obj)(a)6=
∅✳
✷✳ ❈❧❛r❛♠❡♥t❡ s❡ ❝✉♠♣❧❡ ❡st❡ ít❡♠✱ ♣✉❡st♦ q✉❡ ω6∈N0✳
✶✶❱❡r tr❛❜❛❥♦ ❬✶❪ ♣á❣✐♥❛ ✹✳
✶✷P♦r ❝✐❡rt♦ ❞✐r❡♠♦s q✉❡OI(Obj)❡s ✐♥②❡❝t✐✈❛ ❛ ♣❡s❛r ❞❡ ♥♦ s❡ ✉♥❛ ❢✉♥❝✐ó♥✱ ❡❧ ♠♦t✐✈♦ ❞❡ ❡st♦✱ ❡s q✉❡ ❧❛ ♣r♦♣✐❡❞❛❞ ♠♦str❛❞❛ t✐❡♥❡ ❧❛ ❝✉❛❧✐❞❛❞ ❞❡ s❡r✳
✶✻ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
✸✳ ❉❡ ❞❡r❡❝❤❛ ❛ ✐③q✉✐❡r❞❛✳ ▲❛ ❞❡♠♦str❛❝✐ó♥ s❡r❛ ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ ♦r❞❡♥✳ ❙❡❛a6=b✱a, b∈I✱ ❧✉❡❣♦ t❡♥❡♠♦s q✉❡
OI(Obj)(a)∩ OI(Obj)(b) =∅,
♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✶ ② ✷✳ ❆❤♦r❛ s❡❛♥ ordI(a) = m1✱ ②✱
ordI(b) =m2✱ ❝❧❛r❛♠❡♥t❡ s✐m16=m2✱ ❡♥t♦♥❝❡s✱
OI(Obj)(a)∩ OI(Obj)(b) =∅,
❞❛❞❛ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✳ ❙✐ m1 =m2✱ t❡♥❡♠♦s q✉❡ a, b ∈
T(m1)(I)✱ ❡♥ t❛❧ ❝❛s♦✱ a= ha
1, . . . , ani✱ ②✱ b =hb1, . . . , bn′i✳ ❙✐♥
♣❡r❞✐❞❛ ❞❡ ❣❡♥❡r❛❧✐❞❛❞ s✉♣♦♥❣❛♠♦s n≤n′✱ ❧✉❡❣♦ ❞❡❜❡ ♦❝✉rr✐r✿ ❛✮ ❖ ❜✐❡♥ ❡①✐st❡1≤q≤n✱ ❞❡ ♠♦❞♦ q✉❡aq6=bq✳
❜✮ ❖ ❜✐❡♥ ♥♦ ❡①✐st❡ 1 ≤ q ≤ n✱ ❞❡ ♠♦❞♦ q✉❡ aq 6=bq✱ ♣❡r♦
n < n′✳
❊♥ ❡❧ ♣r✐♠❡r ❝❛s♦ ♣♦r ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥ OI(Obj)(aq) ∩
OI(Obj)(bq) =∅✱ ❧✉❡❣♦
OI(Obj)(a1)×. . .×OI(Obj)(an)∩OI(Obj)(b1)×. . .×OI(Obj)(bn′) =∅,
❛sí ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡ P(OI(Obj)(a1)×. . .× OI(Obj)(an)) ❝♦♥ P(OI(Obj)(b1)×. . .× OI(Obj)(bn′))) ={∅}❡s ✈❛❝✐á✱ ❞❡ ❧♦ ❝✉❛❧
❞❡❞✉❝✐♠♦s✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱OI(Obj)(a)∩OI(Obj)(b) =
∅✳ P❛r❛ ❡❧ ❝❛s♦b)✱ ❡s ♦❜✈✐♦ q✉❡ ❧❛ ✐♥t❡rs❡❝❝✐ó♥ ❞❡OI(Obj)(a1)×
. . .×OI(Obj)(an)❝♦♥OI(Obj)(b1)×. . .×OI(Obj)(bn′),❡s ✈❛❝✐á✳
❉❡❧ ♠✐s♠♦ ♠♦❞♦ q✉❡ ❡❧ ♣✉♥t♦ ❛♥t❡r✐♦r ❞❡❞✉❝✐♠♦s
OI(Obj)(a)∩ OI(Obj)(b) =∅.
❉❡ ❞❡r❡❝❤❛ ❛ ✐③q✉✐❡r❞❛✱ s✉♣♦♥❣❛♠♦s q✉❡a=b✱ ❧✉❡❣♦
OI(Obj)(a)∩ OI(Obj)(b) =OI(Obj)(a),
♣❡r♦ ♣♦r ❡❧ ít❡♠ ✶ ❞❡ ❡st❛ ♣r♦♣♦s✐❝✐ó♥OI(Obj)(a)6=∅✱ ❧♦ ❝✉❛❧
♣r♦❞✉❝❡ ✉♥ ❝❧❛r♦ ❝♦♥tr❛s❡♥t✐❞♦ ❝♦♥ ❧❛ ❤✐♣ót❡s✐s✱ ♣♦r ❧♦ t❛♥t♦a6= b✳
✹✳ P♦r ❡❧ ít❡♠ ✶ ② ✸ t❡♥❡♠♦sa6=b❡♥t♦♥❝❡sOI(Obj)(a)6=OI(Obj)(b)✳
▲✉❡❣♦OI(Obj)❡s ✐♥②❡❝t✐✈❛✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✶✼
✺✳ ❙✐ x∈ PI(j)(Obj)✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱ ❡♥t♦♥❝❡s x= (R, j)✱ ❝♦♥ x∈ OI(Obj)(a)✱ ♣❛r❛ ❛❧❣ú♥ a∈ T(j)(I)✳ ❉❡❧ ♠✐s♠♦ ♠♦❞♦
♦❜t❡♥❡♠♦s q✉❡ y = (R′, i)✱ ❝♦♥
y ∈ OI(Obj)(b)✱ ♣❛r❛ ❛❧❣ú♥ b ∈
T(i)(I)✳ ❈❧❛r❛♠❡♥t❡ s✐j6=i❡♥t♦♥❝❡sPI(j)(Obj)∩ PI(i)(Obj) =∅✱
❞❛❞❛ ❧❛ ❝♦♥str✉❝❝✐♦♥❡s ❞❡ ❧♦ ❡❧❡♠❡♥t♦s ❞❛ ❝❛❞❛ ❝♦♥❥✉♥t♦✳ ❉❡ ❞❡r❡❝❤❛ ❛ ✐③q✉✐❡r❞❛✳ ❙✐ P(j)(Obj)∩ P(j)(Obj) = ∅✱ ❡♥t♦♥❡s
♣♦r ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✷ ít❡♠ ✼✱ ② ❧❛s ❝♦♥str✉❝❝✐♦♥❡s ♠♦str❛❞❛s✱ t❡♥❡♠♦si6=j✱ ♦ ❜✐❡♥✱PI(j)(Obj) =∅✳ ❈♦♠♦PI(j)(Obj) =∅❡s ✉♥ ❝♦♥tr❛s❡♥t✐❞♦ ❝♦♥ ❡❧ ít❡♠ ✶✱ ♣✉❡st♦ q✉❡✱ ❞❛❞❛ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✷ ít❡♠ ✻✱ ❡①✐st❡a∈T(j)(I)✱ ❞❡ ♠♦❞♦ q✉❡
∅ 6=OI(Obj)(a)⊂ P
(j)
I (Obj).
P♦r ❧♦ t❛♥t♦j6=i✳
❊♥ ❧♦ q✉❡ s❡ ✈✐❡♥❡ ✉♥❛ ❞❡✜♥✐❝✐ó♥ ❞❡❧ t✐♣♦ ❞❡ ✉♥❛ r❡❧❛❝✐ó♥✱ ❢♦r♠❛❧✐✲ ③❛♥❞♦ ❧♦ ❡①♣✉❡st♦ ❡♥ ❡❧ tr❛❜❛❥♦ ❬✶❪ ♣❛❣✐♥❛ ✺✳❚❛♠❜✐é♥ ♣r❡s❡♥t❛♠♦s ✉♥ ♦♣❡r❛❞♦r q✉❡ ✉s❛r❡♠♦s ❡♥ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✾ ít❡♠ ✹✳
❉❡✜♥✐❝✐ó♥ ✶✳✼✳ ❙❡❛ Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡ I✱ a ∈ TI✱ R ∈
EI(Obj)✳
✶✳ R ❡s ❞❡ t✐♣♦ a ❡♥Obj✱ s✐ ② s♦❧♦ s✐✱R∈ OI(Obj)(a)✳
✷✳ ❙✐ a∈I✱ ❞✐r❡♠♦s q✉❡R ❡s ✉♥ a✲✐♥❞✐✈✐❞✉♦ ❡♥ Obj ✸✳ ❊❧ ♦♣❡r❛❞♦rRel(OI)✱ ❧❛ r❡❧❛❝✐ó♥ ❡♥ OI✱ ❡s ❞❡ ❧❛ ❢♦r♠❛
Rel(OI)(R) =
X ♣❛r❛ R= (X, ordI(a))∈ OI(Obj)(a)
∅ ♣❛r❛ R=∅ω.
◆♦t❡♠♦s q✉❡ ❡❧ ít❡♠ ✶ ❞❡ ❡st❛ ❞❡✜♥✐❝✐ó♥ ❡s ❛❝❡rt❛❞♦ ❞❛❞❛ ❧❛ ♣r♦✲ ♣♦s✐❝✐ó♥ ✶✳✻ ít❡♠ ✸✱ ❡s ❞❡❝✐r✱ ❝❛❞❛ r❡❧❛t♦r t✐❡♥❡ ✉♥ s♦❧♦ t✐♣♦ ❛s♦❝✐❛❞♦✳
❊❧ ♠♦t✐✈♦ ♣❛r❛ ❞❡✜♥✐r ❡❧ ❝♦♥❝❡♣t♦ ❞❡ t✐♣♦ ❞❡ ✉♥❛ r❡❧❛❝✐ó♥ ❞❡ ✉♥❛ ♠❛♥❡r❛ ❛❧❣♦ ❞✐st✐♥t❛ ❞❡ ❧♦ ❤❡❝❤♦ ❡♥ ❬✶❪ ❡s ❡❧✐♠✐♥❛r ❛❧❣✉♥❛s ❛♠❜✐❣ü❡✲ ❞❛❞❡s✳ ❚♦♠❡♠♦s ∅✱ ❡❧ ❝♦♥❥✉♥t♦ ✈❛❝í♦✱ q✉❡ ❡♥ tér♠✐♥♦s ❞❡❧ tr❛❜❛❥♦ ❬✶❪
❝✉♠♣❧❡ ∅ ∈t(hii) ②∅ ∈t(hhiii)✱ ❧✉❡❣♦ t❡♥❡♠♦s✱ ❡♥ tér♠✐♥♦s ❞❡ ❧♦ ❡①✲ ♣✉❡st♦ ❡♥ ❧❛ ♣❛❣✐♥❛ ✹ ② ✺ ❞❡ ❬✶❪✱ q✉❡ ❡❧ ✈❛❝í♦ ❡s ❞❡ t✐♣♦ hii ② hhiii✱
✶✽ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❛❞❡♠ás✱ ❡s ❞❡ ♦r❞❡♥ ✶ ② ✷ ❛❧ ♠✐s♠♦ t✐❡♠♣♦✳ ❊❧ ❝❛♠✐♥♦ q✉❡ s❡ ❤❛ t♦✲ ♠❛❞♦ ♣❛r❛ s♦❧✉❝✐♦♥❛r ❡st♦ ❡s ❝♦♥s✐❞❡r❛r q✉❡ ❧❛ r❡❧❛❝✐ó♥ ✈❛❝í❛ ❡s ✉♥❛ r❡❧❛❝✐ó♥ ❡s♣❡❝✐❛❧ q✉❡ ♥♦ t✐❡♥❡ t✐♣♦✳
◆♦t❡♠♦s q✉❡ s✐ I = {i} r❡❝✉♣❡r❛♠♦s ❧❛ ❞❡✜♥✐❝✐ó♥ ❞❡❧ tr❛❜❛❥♦ ❬✶❪
♣✉❡st♦ q✉❡T =TI✳ ❆✉♥q✉❡ ❝♦♥ ✉♥ ❝✐❡rt♦ r❡♣❛r♦✱ ♣✉❡st♦ q✉❡ ❧❛s r❡❧❛✲
❝✐♦♥❡s ❡♥ ❡st❡ ♠✉♥❞♦ s♦♥ ✉♥ ♣❛r✿ ✉♥ ❝♦♥❥✉♥t♦ r❡❧❛❝✐ó♥ ② s✉ ♦r❞❡♥✳
❊❥❡♠♣❧♦ ✶✳✽✳ ❱❡❛♠♦s✿
✶✳ ❙❡❛ Obj = {N0}✱ s❡❛ I = {i}✱❞♦♥❞❡ ❡s ❝❧❛r♦ q✉❡ Obj ❡s ✉♥❛
❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛✶✸ ❞❡I✱ ② ❡♥ t❛❧ ❝❛s♦✿
❛✮ ❙✐suc={(n, m)∈N0×N0|n=m+1}✱ ❡♥t♦♥❝❡s ❞❡✜♥✐♠♦s
suc1= ({((n,0),(m,0))|(n, m)∈suc},1)∈ OI(Obj)(hi, ii),
❧✉❡❣♦ suc1 ❡s ❞❡ t✐♣♦ hi, ii✳
❜✮ ❙✐+ ={(n1, n2, n3)∈N0×N0×N0|n1+n2=n3}✱ ❡♥t♦♥❝❡s
+1= ({((n1,0),(n2,0),(n3,0))|(n1, n2, n3)∈+},1),
② +1∈ O
I(Obj)(hi, i, ii)✱ ❧✉❡❣♦ +1 ❡s ❞❡ t✐♣♦ hi, i, ii✳
❝✮ ❙✐•={(n1, n2, n3)∈N0×N0×N0|n1•n2=n3}✱ ❡♥t♦♥❝❡s
•1= ({((n
1,0),(n2,0),(n3,0))|(n1, n2, n3)∈+},1),
② •1∈ O
I(Obj)(hi, i, ii)✱ ❧✉❡❣♦ •1 ❡s ❞❡ t✐♣♦ hi, i, ii✳
✷✳ ❙❡❛(G, ⋆)✉♥ ❣r✉♣♦✱ s❡❛Obj={G}✱ s❡❛I={i}✱ ❞♦♥❞❡ ❡s ❝❧❛r♦
q✉❡Obj ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛✶✹ ❞❡I✳ ❊♥ t❛❧ ❝❛s♦✿ ❛✮ ❙✐ ⋆={(g1, g2, g3)∈G×G×G|q1⋆ g2=g4}✱ ❡♥t♦♥❝❡s
⋆1= ({((n1,0),(n2,0),(n3,0))|(n1, n2, n3)∈ •},1),
② ⋆1∈ O
I(Obj)(hi, i, ii)✱ ❧✉❡❣♦ ⋆1 ❡s ❞❡ t✐♣♦ hi, i, ii✳
❜✮ ❙✐eG ❡s ❡❧ ♥❡✉tr♦✱ ❡♥t♦♥❝❡se1G= ({eG},1)❡s ❞❡ t✐♣♦ hii✳
✶✸❙❛t✐s❢❛❝❡ ❧❛s ❝♦♥❞✐❝✐♦♥❡s1.a)✱1.b)②1.c)❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✺✳ ✶✹❙❛t✐s❢❛❝❡ ❧❛s ❝♦♥❞✐❝✐♦♥❡s1.a)✱1.b)②1.c)❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✺✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✶✾
✸✳ ❙❡❛(D,TD)✉♥ ❡s♣❛❝✐♦ t♦♣♦❧ó❣✐❝♦✱ s✉♣♦♥❣❛♠♦s q✉❡D∩P(D) =
∅✳ ◆♦t❡♠♦s q✉❡ Obj ={D,P(D)} ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡
I={i, j}✱ ❞♦♥❞❡ ❡s ❝❧❛r♦ q✉❡Obj ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛✶✺ ❞❡ I✳ ❊♥ t❛❧ ❝❛s♦✿
❛✮ ❙❡❛ ∈D={(a, b)∈D×P(D)|a∈b}✱ ❡♥t♦♥❝❡s
∈1
D= ({((a,0),(b,0)|(a, b)∈∈D},1)∈ OI(Obj)(hi, ji),
❛sí ∈1
D ❡s ❞❡ t✐♣♦ hi, ji✳
❜✮ ❉❡✜♥✐♠♦s
T1D= ({(a,0)|a∈T},1)∈ OI(Obj)(hji),
❧✉❡❣♦ T1D ❡s ❞❡ t✐♣♦ hji✳
✹✳ ❙❡❛(D,UD)✱ ✉♥❛ ✈❛r✐❡❞❛❞ t♦♣♦❧ó❣✐❝❛ ❞❡ ❞✐♠❡♥s✐ó♥n✱ ❝♦♥UD✉♥ ❛t❧❛s ♠❛①✐♠❛❧✳ ❙✉♣♦♥✐❡♥❞♦ q✉❡ Obj={D,P(D),Rn,P(Rn)} ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡I={i, j, o, p}✳ ❊♥t♦♥❝❡s✱ ♣❛r❛ϕ∈UD✱
ϕ⊂A×B✱ ❝♦♥A⊂D ② B⊂Rn✱ ❛❞❡♠ás ❞❡ ♦tr❛s ♣r♦♣✐❡❞❛❞❡s
❡s❡♥❝✐❛❧❡s✱ ♣❡r♦ ❝♦♥ ❧♦ ❞✐❝❤♦ ❡s s✉✜❝✐❡♥t❡ ♣❛r❛ ❞❡✜♥✐r
ϕ1= ({((a,0),(b,0))|(a, b)∈ϕ},1)⊂ OI(Obj)(hi, oi),
❧✉❡❣♦ ϕ1 ❡s ❞❡ t✐♣♦ hi, oi✳ P♦r ❝✐❡rt♦ ♣❛r❛ Obj✱ I ❞❡✜♥✐❞♦ ♠ás ❛rr✐❜❛✱ t❡♥❡♠♦s q✉❡T1D ❡s ❞❡ t✐♣♦hji✱TR1n ❡s ❞❡ t✐♣♦hpi✱∈1D ❡s
❞❡ t✐♣♦hi, oi✱ ②✱∈1
Rn ❡s ❞❡ t✐♣♦ ho, pi✳
❉❡✜♥✐❝✐ó♥ ✶✳✾✳ ❙❡❛Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡I✳
✶✳ ❯♥❛ ♣r❡✲❡str✉❝t✉r❛ ❡♥ Obj✱ ❡s ✉♥❛ ❢✉♥❝✐ó♥ Ψ :N0∪ {ω} 7→ EI(Obj)∪ {∅ω}✱ q✉❡ ❝✉♠♣❧❡
Ψ(l)⊂ PI(l)(Obj)✱ ♣❛r❛l∈N0✱ ②✱Ψ(ω) =∅ω.
✷✳ ❙❡❛ a ∈ TI✱ ❞❡✜♥✐♠♦s ❡❧ ✉♥✐✈❡rs♦ ❞❡ t✐♣♦ a ❞❡ Ψ ❝♦♠♦
Ua(Ψ) = Ψ(ord
I(a))∩ OI(Obj)(a)✳
✸✳ ❊❧ ✉♥✐✈❡rs♦ ❞❡ Ψ✱ U(Ψ)✱ ❝♦♠♦ U(Ψ) =SΨ[N0]✱ ❞♦♥❞❡ Ψ[ ]✱
❡s ❡❧ ♦♣❡r❛❞♦r ✐♠❛❣❡♥ ❞❡ Ψ✳
✶✺❙❛t✐s❢❛❝❡ ❧❛s ❝♦♥❞✐❝✐♦♥❡s1.a)✱1.b)②1.c)❞❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ✶✳✺✳
✷✵ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
✹✳ Ψ❡s ✉♥❛ ❡str✉❝t✉r❛ ❡♥Obj✱ s✐ ② s♦❧♦ s✐✿ ❡s ✉♥❛ ♣r❡✲❡str✉❝t✉r❛✱ ② ♣❛r❛ ❝❛❞❛R∈U(Ψ)❞❡ t✐♣♦ a=ha1, . . . , ani❡♥ TI
Rel(OI)(R)⊂Ua1(Ψ)×. . .×Uan(Ψ).
P♦r ❧♦ ❣❡♥❡r❛❧ ❞❡♥♦t❛r❡♠♦s ❧❛s ❡str✉❝t✉r❛s ❝♦♠♦ s✉❝❡s✐♦♥❡s ❡♥N0✱
♣✉❡st♦ q✉❡ s✉ ✉❧t✐♠♦ ✈❛❧♦r ❡s ❡❧ ♠✐s♠♦ ♣❛r❛ ❝✉❛❧q✉✐❡r❛ ② ❡s ✐rr❡❧❡✈❛♥t❡ ♣❛r❛ ❧❛ ❞✐❢❡r❡♥❝✐❛❝✐ó♥ ❡♥tr❡ ❡❧❧❛s✳
P✉❡❞❡ q✉❡ ❧❛ ❞❡✜♥✐❝✐ó♥ ♣❛r❡③❝❛ ♣♦❝♦ ✐♥t✉✐t✐✈❛ ② ❞✐✜❡r❛ ❞❡ ❧♦ ♠♦s✲ tr❛❞♦ ♣♦r ◆❡✇t♦♥ ❞❛ ❈♦st❛ ❡♥ ❬✶❪✱ s✐♥ ❡♠❜❛r❣♦ ést❛ ❡s ❝❛♣❛③ ❞❡ ♠♦❞❡❧❛r ❜❛st❛♥t❡ ❜✐❡♥ ❧❛s ❡str✉❝t✉r❛s q✉❡ é❧ ♣❧❛♥t❡❛✳ P❛r❛ ❡❧❧♦ t♦♠❡♠♦s ❧❛ ❡s✲ tr✉❝t✉r❛ e = hD, rli✱ ❡♥ ❡❧ s❡♥t✐❞♦ ❞❡ ❬✶❪✱ ❧♦ ♣r✐♠❡r♦ q✉❡ ❤❛r❡♠♦s ❡s
♥♦♠❜r❛r Rl ❛❧ ❝♦♥❥✉♥t♦ ❞❡ ❧♦s ❡❧❡♠❡♥t♦s ❞❡rl✳ ▲✉❡❣♦✱ ✜❥❛♥❞♦ I={i}
② Obj ={D}✱ tr❛♥s❢♦r♠❛♠♦s ❧❛s r❡❧❛❝✐♦♥❡s ❞❡ Rl ❝♦♠♦ ❤❡♠♦s ♠❡♥✲
❝✐♦♥❛❞♦ ❡♥ ❡❧ ♣árr❛❢♦ ❛♥t❡❝❡❞❡♥t❡ ❛❧ ❡❥❡♠♣❧♦ ✶✳✽✳ ❊❧ ❝♦♥❥✉♥t♦ ❞❡ ❧♦s ❡❧❡♠❡♥t♦s tr❛♥s❢♦r♠❛❞♦s ❧♦ ❞❡♥♦t❛r❡♠♦s ♣♦r R∗l✳ ❨❛ ❡st❛♠♦s ❝❛s✐ ❧✐s✲ t♦s✱ ♣❛r❛ t❡r♠✐♥❛r ❝♦♥str✉✐♠♦s ❧❛ s✉❝❡s✐ó♥
hD× {0}, R∗l ∩ PI(1)(Obj), Rl∗∩ P
(2)
I (Obj), . . . , R
∗
l ∩ P
(n)
I (Obj), . . .i,
q✉❡ ❡s ❡❧ ❛♥á❧♦❣♦ ❞❡ ❧❛ ❡str✉❝t✉r❛ ❞❡ ◆❡✇t♦♥ ❞❛ ❈♦st❛✶✻❡♥ ❡st❛ ❝♦♥s✲
tr✉❝❝✐ó♥✳
P✉❡❞❡ r❡s✉❧t❛r ❝✉r✐♦s♦ ❡❧ ❤❡❝❤♦ q✉❡ s❡ ❞❡✜♥✐❡r❛ ♣r❡✲❡str✉❝t✉r❛ ② ♥♦ ❞✐r❡❝t❛♠❡♥t❡ ❡str✉❝t✉r❛✳ ❊❧ ♠♦t✐✈♦ ❞❡ ❡❧❧♦ ❡s q✉❡ ❧❛ ♣r❡✲❡str✉❝t✉r❛ ♣♦s❡❡ ❛❧❣✉♥♦s ❝❛s♦s ❛♥ó♠❛❧♦s✳ P♦r ❡❥❡♠♣❧♦ ❧❛ s✐❣✉✐❡♥t❡ ♣r❡✲❡str✉❝t✉r❛✿
hR0,{+1C,·1C},∅, . . .i✱ ❞♦♥❞❡R0 =R× {0} ✱ ②+1C,·1C s♦♥ ❧❛ s✉♠❛ ② ❡❧
♣r♦❞✉❝t♦s ❡♥ ❧♦s ❝♦♠♣❧❡❥♦s ♠♦❞✐✜❝❛❞♦s ♣❛r❛ ❡st❛r ❛❝♦r❞❡ ❛ ❧❛ ❞❡✜♥✐❝✐ó♥ ❛♥t❡r✐♦r✳
❉❡✜♥✐❝✐ó♥ ✶✳✶✵✳ ❙❡❛ Ψ✉♥❛ ♣r❡✲❡str✉❝t✉r❛ ❡♥ Obj✱ ❝♦♥ Obj ✉♥❛ ❝♦✲ ❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡I✳ ❉❡✜♥✐♠♦s✿
✶✳ P❛r❛R∈U(Ψ)✿
ordI(R) =ordI(a)✱ s✐R∈ OI(Obj)(a)♣❛r❛ ❛❧❣ú♥a∈TI.
▲❧❛♠❛r❡♠♦s ❛ ordI(R)✱ ❡❧ ♦r❞❡♥ ❞❡ R✳
✶✻❱❡r tr❛❜❛❥♦ ❬✶❪ ♣á❣✐♥❛ ✹✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✷✶
✷✳ ❊❧ ♦r❞❡♥ ❞❡ Ψ❡s ❞❡ ❧❛ ❢♦r♠❛✿
ordI(Ψ) =
m´ax{ordI(R)|R∈U(Ψ)} ✱ s✐ ❡st❡max❡①✐st❡
ω ✱ ❡♥ ♦tr♦ ❝❛s♦.
◆♦t❡♠♦s q✉❡ ❡st❛ ❞❡✜♥✐❝✐ó♥ ❡stá ❜✐❡♥ ❢✉♥❞❛❞❛ ❞❛❞❛ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✻ ít❡♠ ✷ ② ✸✳ ❊st♦ ♥♦s ❛s❡❣✉r❛ q✉❡ ♥✐♥❣ú♥ r❡❧❛t♦r t❡♥❞rá ♠❛s ❞❡ ✉♥ t✐♣♦❀ ❞❡❧ ♠✐s♠♦ ♠♦❞♦✱ ❡❧ ♦r❞❡♥ ❛s♦❝✐❛❞♦ ❡s ú♥✐❝♦✳
✷✳ ❊❥❡♠♣❧♦s ❞❡ ❡str✉❝t✉r❛s
❊str✉❝t✉r❛ s✐♠♣❧❡ ② ♠ú❧t✐♣❧❡
❍❡♠♦s ✈✐st♦ ❡♥ ✈❛r✐♦s ❡❥❡♠♣❧♦s q✉❡ ❡❧ ❝❛s♦ ♣❛rt✐❝✉❧❛r ❞❡ ❡str✉❝t✉r❛s ❡♥ ❡❧ ❝✉❛❧ s❡ ✜❥❛I={i}②Obj ={D}✱ ♣❛r❛D✉♥ ❝♦♥❥✉♥t♦ ❝✉❛❧q✉✐❡r❛✱ ❡s ♠✉② r❡❝✉rr❡♥t❡ ❛❧ ♠♦♠❡♥t♦ ❞❡ tr❛t❛r ❞❡ ❡♥t❡♥❞❡r ❛ ◆❡✇t♦♥ ❞❛ ❈♦st❛✱ t❛❧ ❤❡❝❤♦ ❡s ❡❧ q✉❡ ♠♦t✐✈❛ ❧❛ s✐❣✉✐❡♥t❡ ❞❡✜♥✐❝✐ó♥✳
❉❡✜♥✐❝✐ó♥ ✷✳✶✳ ❙❡❛Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ❞❡I✳ ❉✐r❡♠♦s q✉❡ ✶✳ ❙✐Obj ={D}✱ ❝♦♥D ✉♥ ❝♦♥❥✉♥t♦ ❝✉❛❧q✉✐❡r❛✱ ✉♥❛ ❡str✉❝t✉r❛ ❡♥
Obj s❡rá ❞❡♥♦t❛❞❛ ♣♦re✱ ② s❡rá ❧❧❛♠❛❞❛ ❡str✉❝t✉r❛ s✐♠♣❧❡✳ ✷✳ ❙✐#Obj >1✱ ❞♦♥❞❡#❡s ❡❧ ♦♣❡r❛❞♦r ❝❛r❞✐♥❛❧✱ ✉♥❛ ❡str✉❝t✉r❛ ❡♥
Obj s❡rá ❞❡♥♦t❛❞❛ ♣♦rµ✱ ② s❡rá ❧❧❛♠❛❞❛ ❡str✉❝t✉r❛ ♠ú❧t✐♣❧❡✳
▲❛s ❡str✉❝t✉r❛s s✐♠♣❧❡s s♦♥ ♥✉❡str♦ ❛♥á❧♦❣♦ ❛ ◆❡✇t♦♥ ❞❛ ❈♦st❛✳ ▲❛s ❡str✉❝t✉r❛s ♠ú❧t✐♣❧❡s ♥♦s ♣❡r♠✐t✐rá✱ ❤❛❜❧❛r ❞❡ ❡str✉❝t✉r❛s ❞❡ ✈❛✲ r✐♦s ✐♥❞✐✈✐❞✉♦s✳ ❙✐♥ ❡♠❜❛r❣♦✱ s❡❣ú♥ ◆❡✇t♦♥ ❞❛ ❈♦st❛✱ s♦♥ ♣r❡s❝✐♥❞✐❜❧❡s✱ ♣✉❡st♦ q✉❡ s❡ ♣✉❡❞❡♥ r❡❞✉❝✐r ❛ ✉♥❛ ❡str✉❝t✉r❛ s✐♠♣❧❡✱ ♣❡r♦ ❝♦♠♦ ♥♦ s❡ ❤❛ ❞❡♠♦str❛❞♦ t❛❧ ❛✜r♠❛❝✐ó♥ tr❛❜❛❥❛ré ❝♦♥ ❡❧❧❛s✳ ❊♥ ♠✐ ♦♣✐♥✐ó♥ ❧❛s ❡s✲ tr✉❝t✉r❛s ♠ú❧t✐♣❧❡s s✐♠♣❧✐✜❝❛♥ ❧❛ tr❛❞✉❝❝✐ó♥ ❞❡ ❝♦♥❝❡♣t♦s ♠❛t❡♠át✐❝♦s ❛ ❡str✉❝t✉r❛s✱ ❝♦♠♦ ✈✐♠♦s ❡♥ ❡❧ ❡❥❡♠♣❧♦ ✶✳✽✳
❊❥❡♠♣❧♦ ✷✳✷✳ ❯s❛♥❞♦ ❡❧ ❡❥❡♠♣❧♦ ✶✳✽ ✈❡❛♠♦s ✶✳ ❙❡❛Obj={N0}✱ ❞❡✜♥✐♠♦se❝♦♠♦✿
❛✮ e(0) =N0× {0}✳
❜✮ e(1) ={+1,•1, suc1}✳
✷✷ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
❝✮ e(l) =∅✱ ♣❛r❛1< l < ω✳
❊st❛ ❡str✉❝t✉r❛ ♣❡r♠✐t❡ ❞❡ ♠♦❞❡❧❛r ❧❛ ❛r✐t♠ét✐❝❛✳ ●❡♥❡r❛❧♠❡♥t❡ s❡ ❞❡♥♦t❛ ♣♦r
e=hN0× {0},{+1,•1, suc1}i.
✷✳ ❙❡❛Obj={G}✱ ❞❡✜♥✐♠♦se❝♦♠♦✿ ❛✮ e(0) =G× {0}✳
❜✮ e(1) ={⋆1, e1G}✳
❝✮ e(l) =∅✱ ♣❛r❛1< l < ω✳
❊st❛ ❡str✉❝t✉r❛ ♣❡r♠✐t❡ ❞❡ ♠♦❞❡❧❛r ❧❛ ♥♦❝✐ó♥ ❞❡ ❣r✉♣♦✳ ●❡♥❡r❛❧✲ ♠❡♥t❡ s❡ ❞❡♥♦t❛ ♣♦r
e=hG× {0},{⋆1, e1G}i. ✸✳ ❙❡❛Obj={D,P(D)}✱ ❞❡✜♥✐♠♦sµ❝♦♠♦✿
❛✮ µ(0) = (D∪P(D))× {0}✳
❜✮ µ(1) ={T1D,∈1
D}✳
❝✮ µ(l) =∅✱ ♣❛r❛1< l < ω✳
❊st❛ ❡str✉❝t✉r❛ ♣❡r♠✐t❡ ❞❡ ♠♦❞❡❧❛r ❧❛ ♥♦❝✐ó♥ ❞❡ t♦♣♦❧♦❣í❛✳ ●❡✲ ♥❡r❛❧♠❡♥t❡ s❡ ❞❡♥♦t❛ ♣♦r
µ=h(D∪P(D))× {0},{T1D,∈1
D}i.
✹✳ ❙❡❛Obj={D,P(D),Rn,P(Rn)}✱ ❞❡✜♥✐♠♦s µ❝♦♠♦✿
❛✮ µ(0) = (D∪P(D)∪Rn∪P(Rn))× {0}✳
❜✮ µ(1) = {T1D,∈1
D,T1Rn,∈1Rn} ∪U∗D✱ ❞♦♥❞❡ U∗D = {ϕ1|ϕ ∈
UD}
❝✮ µ(l) =∅✱ ♣❛r❛1< l < ω✳
❊st❛ ❡str✉❝t✉r❛ ♣❡r♠✐t❡ ❞❡ ♠♦❞❡❧❛r ❧❛ ♥♦❝✐ó♥ ❞❡ ✈❛r✐❡❞❛❞ t♦♣♦✲ ❧ó❣✐❝❛✳ ●❡♥❡r❛❧♠❡♥t❡ s❡ ❞❡♥♦t❛ ♣♦r
µ=h(D∪P(D)∪Rn∪P(Rn))× {0},{T1D,∈1
D,T1Rn,∈1Rn} ∪U ∗
Di.
❊♥ ❧♦ q✉❡ s✐❣✉❡ s✐I ={i}✱ ❡♥t♦♥❝❡sTI s❡rá ❞❡♥♦t❛❞♦ ♣♦rT✱ordI
♣♦rord✱ OI ♣♦rO✱EI ♣♦rE✱PI(l)♣♦rP(l)✱ ❝♦♠♦ ❡♥ ❡❧ ❛rt✐❝✉❧♦ ❬✶❪✳
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✷✸
❊str✉❝t✉r❛ ❛♥á❧♦❣❛
❆ ❝♦♥t✐♥✉❛❝✐ó♥ ❝♦♥str✉✐r❡♠♦s ❧❛s ❤❡rr❛♠✐❡♥t❛s ♥❡❝❡s❛r✐❛s ♣❛r❛ ♣♦✲ ❞❡r r❡❞✉❝✐r ❡str✉❝t✉r❛s ♠ú❧t✐♣❧❡s ❛ ❡str✉❝t✉r❛s s✐♠♣❧❡s✳ P❛r❛ ❡❧❧♦ ❝♦♥s✲ tr✉✐r❡♠♦s ❧❛s tr❛♥s❢♦r♠❛❝✐♦♥❡s ❞❡ t✐♣♦s ❝♦♥✈❡♥✐❡♥t❡s✳ ❉❡s♣✉és ❝♦♥str✉✐✲ r❡♠♦s ❝♦❧❡❝❝✐♦♥❡s ♦❜❥❡t✐✈❛s ❝♦♥✈❡♥✐❡♥t❡s ♣❛r❛ ❡st❛s tr❛♥s❢♦r♠❛❝✐♦♥❡s✱ ② ♣♦r ú❧t✐♠♦ ❝♦♥str✉✐r❡♠♦s ❧❛ ❡str✉❝t✉r❛ q✉❡ ♣❡r♠✐t✐rá ❧❛ r❡❞✉❝❝✐ó♥✳ ❉❡✜♥✐❝✐ó♥ ✷✳✸✳ ❙❡❛J ✉♥ ❝♦♥❥✉♥t♦ ❛r❜✐tr❛r✐♦ ❞♦♥❞❡TJ ❡s ✉♥ ❝♦♥❥✉♥t♦
❞❡ t✐♣♦s ② s❡❛ I={i} ❝♦♠♦ ❛rr✐❜❛✳ ❉❡✜♥✐♠♦s✱ ❧❛ ❢✉♥❝✐ó♥ s✐♥❣✉❧❛r ❞❡ J✱ ❝♦♠♦ ✉♥❛ ❢✉♥❝✐ó♥SJ :TJ7→T✱ q✉❡ ❝✉♠♣❧❡
✶✳ SJ(a) =i✱ ♣❛r❛ t♦❞♦ a∈J✳
✷✳ s✐a=ha1, . . . , ani✱ ❡♥t♦♥❝❡sSJ(a) =hSJ(a1), . . . ,SJ(an)i✳
Pr♦♣♦s✐❝✐ó♥ ✷✳✹✳ ❙❡❛ # ❡s ❡❧ ♦♣❡r❛❞♦r ❝❛r❞✐♥❛❧✳ ❙❡ ❝✉♠♣❧❡ q✉❡✿ ✶✳ ♣❛r❛ t♦❞♦ a∈TJ✱ordJ(a) =ord(SJ(a))✳
✷✳ SJ ❡s s♦❜r❡②❡❝t✐✈❛✳
✸✳ ❙❡❛a∈T✱ s❡ ❝✉♠♣❧❡ q✉❡✿
#SJ[{a}]≤ω✱ s✐ ② s♦❧♦ s✐✱ #J ≤ω.
❉❡♠♦str❛❝✐ó♥✳ ❊♥✉♠❡r❛♥❞♦ ❧❛s ❞❡♠♦str❛❝✐♦♥❡s✿
✶✳ ▲❛ ❞❡♠♦str❛❝✐ó♥ s❡rá ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ ♦r❞❡♥ ❞❡ a ∈ TJ✱
s❡❛ ordJ(a) = 0✱ ❧✉❡❣♦ ♥❡❝❡s❛r✐❛♠❡♥t❡ a ∈ J✱ ❞❡ ❧♦ ❝♦♥tr❛r✐♦
♣♦r ♣r♦♣♦s✐❝✐ó♥ ✶✳✹ ít❡♠ ✷ ordI(a)≥1✳ ❆sía∈J ② SJ(a) =i✱
❧✉❡❣♦ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✸ ord(SJ(a)) = 0✳ P♦r ❧♦ t❛♥t♦ordJ(a) =
ord(SI(a))✳
P❛r❛ordJ(a)≥1✱ t❡♥❡♠♦s q✉❡a=ha1, . . . , ani✱ ❞❡ ❧♦ ❝♦♥tr❛r✐♦
a ∈J ❧♦ ❝✉❛❧ ❡s ✉♥❛ ❝❧❛r❛ ❝♦♥tr❛❞✐❝❝✐ó♥ ❝♦♥ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✹ ít❡♠ ✶✳ ▲✉❡❣♦✿
ordJ(a) = m´ax{ordJ(a1), . . . , ordJ(an)}+ 1,
❝♦♠♦ ordJ(aj) < ordJ(a)✱ ♣❛r❛ 1 ≤ j ≤ n✱ ♣♦r ❤✐♣ót❡s✐s ❞❡
✐♥❞✉❝❝✐ó♥
ordJ(a) = m´ax{ord(SJ(a1)), . . . , ord(SJ(an))}+ 1,
❞❡ ❧♦ ❝✉❛❧ s❡ ❞❡❞✉❝❡✱ ♣♦r ♣r♦♣♦s✐❝✐ó♥ ✶✳✹✱ordJ(a) =ord(SJ(a))✳
✷✹ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
✷✳ ❉❡♠♦str❛r❡♠♦s ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ ♦r❞❡♥✳ ❙❡❛a∈T❞❡ ♠♦❞♦ q✉❡ ord(a) = 0✱ ❧✉❡❣♦ ♥❡❝❡s❛r✐❛♠❡♥t❡ a=i✱ ❞❡ ❧♦ ❝♦♥tr❛r✐♦ t❡✲ ♥❡♠♦s ✉♥ ❝♦♥tr❛s❡♥t✐❞♦ ❝♦♥ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✹✳ ❆❤♦r❛ ❡s❝♦❥❛♠♦s ❝✉❛❧q✉✐❡r j∈J ♣❛r❛ ♦❜t❡♥❡r ♣♦r ❡❧ ít❡♠ ✶ ② ❞❡✜♥✐❝✐ó♥ ✷✳✸
ordJ(j) =ord(SJ(j)) =ord(i) = 0,
❧♦ ❝✉❛❧ ❛✜r♠❛ q✉❡ SJ(j) =i✳
P❛r❛ord(a)≥1✱ t❡♥❡♠♦s q✉❡ a=ha1, . . . , ani✱ ♣❛r❛ 1≤n < ω✱
❧✉❡❣♦✱ ♣♦r ♣r♦♣♦s✐❝✐ó♥ ✶✳✹
ord(a) = m´ax{ord(a1), . . . , ord(an)}+ 1,
✉s❛♥❞♦ ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥✱ ②❛ q✉❡ ord(al) < ord(a) ♣❛r❛
1≤l≤n✱ ❡①✐st❡ bl∈TJ✱ t❛❧ q✉❡SJ(bl) =al✳ ❆sí ♣♦r ❞❡✜♥✐❝✐ó♥
✷✳✸
a=hSJ(a1), . . . ,SJ(an)i=SJ(hb1, . . . , bni).
P♦r ❧♦ t❛♥t♦SI ❡s s♦❜r❡②❡❝t✐✈❛✳
✸✳ ❙❡❛ SJ−1 ❡s ❧❛ ✐♠❛❣❡♥ ✐♥✈❡rs❛ ❞❡❧ ❝♦♥❥✉♥t♦ ❜❛❥♦ SJ✳ ▲❛ ❞❡♠♦s✲
tr❛❝✐ó♥ s❡r❛ ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ord(a)✳
• P❛r❛ ord(a) = 0 t❡♥❡♠♦s q✉❡ a = i ② SJ[{i}] = J✱ ❛sí
❝❧❛r❛♠❡♥t❡#S−1
J [{i}]≤ω✱ s✐ ② s♦❧♦ s✐#J ≤ω✳
• ❙✐ ord(a) > 0✱ ❧✉❡❣♦ a = ha1, . . . , ani✱ ♣❛r❛ ❛❧❣ú♥ n ∈
N✳ ❆❤♦r❛ ✈❡❛♠♦s q✉❡ s✐ a′ ∈ S−1
J [{a}]✱ ❡♥t♦♥❝❡s a
′ =
ha′1, . . . , a′mi✱ ♣❛r❛ ❛❧❣ú♥ m ∈ N✱ ❡st♦ ✈✐❡♥❡ ❞❛❞♦ ♣♦r ❡❧
ít❡♠ ✶ ❞❡ ❡st❛ ♣r♦♣♦s✐❝✐ó♥ ② ❡❧ ❝♦♥tr❛s❡♥t✐❞♦ q✉❡ s❡ ♣r♦✲ ❞✉❝❡ ❝♦♥ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✶✳✹ s✐ m = 0✳ ▼❛s ❛ú♥✱ ♣♦❞❡♠♦s ❞❡❞✉❝✐r q✉❡ m = n ② SJ(aj) = a′j✱ ♣❛r❛ 1 ≤ j ≤ n✳ ▲♦
q✉❡ ❡q✉✐✈❛❧❡ a′j ∈ S−1
J [{aj}]✱ ♣❛r❛ 1 ≤j ≤n✳ ❈♦♠♦ a′ ❡s
❛r❜✐tr❛r✐♦ ❡s ❛r❜✐tr❛r✐♦ t❡♥❡♠♦s q✉❡
#SJ−1[{a}] = #(SJ−1[{a1}]×. . .× SJ−1[{an}]).
P♦r ❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥✱ ②❛ q✉❡ ♣♦r ♣r♦♣♦s✐❝✐ó♥ ✶✳✹✱ ord(aj)< ord(a)♣❛r❛1≤j≤n✱ t❡♥❡♠♦s q✉❡✿
#SJ−1[{aj}]≤ω✱ s✐ ② s♦❧♦ s✐#J ≤ω✳ ❈♦♠♦SJ−1[{a}]≤ω✱
❡s ❡q✉✐✈❛❧❡♥t❡ ❛ #SJ−1[{aj}] ≤ω✱ ♣❛r❛1 ≤ j ≤n✳ P♦r ❧♦
P❘❖▲❊●Ó▼❊◆❖❙ P❆❘❆ ❯◆❆ ❚❊❖❘❮❆ ❋❖❘▼❆▲ ❉❊ ❊❙❚❘❯❈❚❯❘❆❙ ✷✺
P❛r❛ ❧♦ q✉❡ s❡ ✈✐❡♥❡✱ s❡❛Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ♣❛r❛I❛r❜✐tr❛✲ r✐♦✳ ❊♥t♦♥❝❡s✱ ♥♦t❡♠♦s q✉❡ Obji ={SObj} ❡s ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛
❞❡ {i}✳ ❯s❛♠♦s Obji ♣❛r❛ ❝r❡❛r ✉♥❛ ❡str✉❝t✉r❛ ❝♦♥✈❡♥✐❡♥t❡ ♣❛r❛ ❧❛
r❡❞✉❝❝✐ó♥ ❞❡ ✐♥❞✐✈✐❞✉♦s✳
Pr♦♣♦s✐❝✐ó♥ ✷✳✺✳ ❙❡❛♥✿Obj ✉♥❛ ❝♦❧❡❝❝✐ó♥ ♦❜❥❡t✐✈❛ ♣❛r❛I✱Obji ❝♦♠♦
❛rr✐❜❛✱ l∈N0✱ ②a∈TI✳
✶✳ S{O
I(Obj)(a)| ∈I}=O(Obji)(i)✱
❡s ❞❡❝✐rPI(0)(Obj) =P(0)(Obj
i).
✷✳ OI(Obj)(a)⊂ O(Obji)(SI(a))✳
✸✳ R⊂ OI(Obj)(a)✱ ②✱R6=∅✱ ❡♥t♦♥❝❡s(R, ordI(a) + 1)❡s ❞❡ t✐♣♦
hSI(a)i ❡♥Obji✳
✹✳ s✐R ❡s ❞❡ t✐♣♦ a❡♥Obj✱ ❡♥t♦♥❝❡sR ❡s ❞❡ t✐♣♦ SI(a) ❡♥Obji✳
✺✳ s✐ R ❡s ✉♥ a✲✐♥❞✐✈✐❞✉♦ ❡♥ Obj ❡♥t♦♥❝❡s R ❡s ✉♥ ✐♥❞✐✈✐❞✉♦ ❡♥ Obji✳
✻✳ PI(l)(Obj)⊂ P(l)(Obj
i)✳
✼✳ EI(Obj)⊂ E(Obji)✳
✽✳ R∈ PI(l)(Obj)✱ ♣❛r❛ ❛❧❣ú♥l∈N✱
❡♥t♦♥❝❡s Rel(OI)(R) =Rel(O)(R)✳
❉❡♠♦str❛❝✐ó♥✳ ❊♥✉♠❡r❛♥❞♦ ❧❛s ❞❡♠♦str❛❝✐♦♥❡s✿ ✶✳ ❱❡❛♠♦s q✉❡✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱
[
{OI(Obj)(a)|a∈I}= (
[
{f(a)|a∈I})× {0}=O(Obji)(i),
❞♦♥❞❡ f ❡s ❧❛ ❜✐②❡❝❝✐ó♥ ❞❡ I ❡♥Obj✳ ❈❧❛r❛♠❡♥t❡ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷ ② ❧♦ ♠❡♥❝✐♦♥❛❞♦✱
PI(0)(Obj) =P(0)(Obji).
✷✻ ❱■❉❆▲ ❍❯▼❇❊❘❚❖ ◆❆❱❆❘❘❖ ▼❊◆❆
✷✳ ▲❛ ❞❡♠♦str❛❝✐ó♥ s❡r❛ ♣♦r ✐♥❞✉❝❝✐ó♥ s♦❜r❡ ❡❧ ♦r❞❡♥ ❞❡ a ∈ TI✳
P❛r❛ ordI(a) = 0✱ ❡♥t♦♥❝❡s a ∈ I✱ ❧✉❡❣♦ ♣♦r ❡❧ ít❡♠ ✶ ❞❡ ❡st❛
♣r♦♣♦s✐❝✐ó♥ ② ❧❛ ❞❡✜♥✐❝✐ó♥ ✷✳✸✱
OI(Obj)(a)⊂ O(Obj)(i)(SI(a)).
P❛r❛ordI(a)≥1✱ t❡♥❡♠♦sa=ha1, . . . , ani✳ ❯s❛♥❞♦ ❧❛ ♣r♦♣♦s✐✲
❝✐ó♥ ✶✳✹✱ t❡♥❡♠♦sordI(aj)< ordI(a)✱ ♣❛r❛1≤j ≤n✱ ❧✉❡❣♦ ♣♦r
❤✐♣ót❡s✐s ❞❡ ✐♥❞✉❝❝✐ó♥✱ ❧❛ ♣r♦♣♦s✐❝✐ó♥ ✷✳✹ ít❡♠ ✶ ② ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱
OI(Obj)(a)⊂(P(O(Obji)(a1)×. . .×O(Obji)(a)))×{ord(SI(a))}.
P♦r ❧♦ t❛♥t♦✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱
OI(Obj)(a)⊂ O(Obji)(SI(a)).
✸✳ ◆♦t❡♠♦s q✉❡ ♣♦r ❤✐♣ót❡s✐s ❡ ít❡♠ ❛♥t❡r✐♦r t❡♥❡♠♦s✱ ♣❛r❛ t♦❞♦ a∈TI✱
R∈P(OI(Obj)(a))\{∅} ⊂P(O(Obji)(SI(a)))\{∅},
❧✉❡❣♦ t❡♥❡♠♦s q✉❡
(R, ordI(a) + 1)∈(P(O(Obji)(SI(a)))\{∅})× {ordI(a) + 1}.
P❡♥s❛♥❞♦ ❡♥ ❧❛ ♣❡rt❡♥❡♥❝✐❛ ♠♦str❛❞❛✱ ✈❡❛♠♦s q✉❡✱ ♣♦r ♣r♦♣♦s✐✲ ❝✐ó♥ ✷✳✹ ít❡♠ ✶✱
ordI(hai) =ordI(a) + 1 =ordI(SI(a)) + 1 =ord(SI(a)).
▲✉❡❣♦✱ ♣♦r ❞❡✜♥✐❝✐ó♥ ✶✳✺ ít❡♠ ✷✱
(R, ordI(a) + 1)∈ O(Obji)(SI(a)).
P♦r ❧♦ t❛♥t♦(R, ordI(a) + 1)❡s ❞❡ t✐♣♦SI(a)❡♥Obji✳
✹✳ ❈♦♥s❡❝✉❡♥❝✐❛ ❞✐r❡❝t❛ ❞❡❧ ít❡♠ ✷✱ ♣✉❡st♦ q✉❡✿R❞❡ t✐♣♦a❡♥Obj✱ ❡♥t♦♥❝❡s R ∈ OI(Obj)(a)✱ ❧✉❡❣♦ R ∈ O(Obji)(SI(a))✱ ❛sí R ❡s
❞❡ t✐♣♦SI(a)❡♥Obji✳
