U matematici osoblivo v teoriyi mnozhin kazhut sho dvi vporyadkovani mnozhini X i Y mayut odnakovij tip poryadku yaksho voni poryadkovo izomorfni tobto yaksho isnuye biyekciya kozhen element vidpovidaye rivno odnomu elementu z inshoyi mnozhini f X Y displaystyle f colon X to Y taka sho i f i obernena do neyi monotonni zberigayut poryadok elementiv V osoblivomu vipadku koli X linijno vporyadkovana monotonnist f tyagne za soboyu monotonnist yiyi obernennya Napriklad mnozhina vsih cilih i mnozhina parnih cilih mayut odnakovij tip poryadku bo vidobrazhennya n 2 n displaystyle n mapsto 2n ce biyekciya zi zberezhennyam poryadku Ale tipi poryadku mnozhini cilih i mnozhini racionalnih zi standartnim poryadkom vidriznyayutsya bo navit nezvazhayuchi na te sho obidvi mnozhini mayut odnakovu potuzhnist voni zlichenno neskinchenni mizh nimi ne isnuye vidobrazhennya zi zberezhennyam poryadku Inshij priklad mnozhin iz odnakovim tipom poryadku mozhe buti dodatni cili mayut najmenshij element i vid yemni cili mayut najbilshij element Vidkritij interval 0 1 racionalnih chisel poryadkovo izomorfnij vsim racionalnim bo napriklad f x 2 x 1 1 2 x 1 displaystyle f x tfrac 2x 1 1 vert 2x 1 vert strogo zrostna biyekciya z pershoyi na drugu mnozhinu Cherez te sho rivnist poryadkiv ce vidnoshennya ekvivalentnosti vona rozbivaye klas vsih vporyadkovanih mnozhin na klasi ekvivalentnosti Zmist 1 Tip poryadku cilkovitih vporyadkuvan 2 Kozhna cilkom vporyadkovana mnozhina maye tip poryadku 2 1 Dovedennya 3 PosilannyaTip poryadku cilkovitih vporyadkuvan RedaguvatiKozhna cilkom vporyadkovana mnozhina poryadkovo izomorfna rivno odnomu poryadkovomu chislu i cej izomorfizm unikalnij Teper mi gotovi dati take oznachennya Tip poryadku cilkom vporyadkovanoyi mnozhini S displaystyle S leq nbsp ce unikalne poryadkove chislo yake poryadkovo izomorfno S displaystyle S leq nbsp Tip poryadku chasto poznachayut yak Ord S displaystyle operatorname Ord S leq nbsp Kozhna cilkom vporyadkovana mnozhina maye tip poryadku RedaguvatiShob dovesti sho kozhna cilkom vporyadkovana mnozhina maye tip poryadku nam dostatno dovesti sho kozhna cilkom vporyadkovana mnozhina poryadkovo izmorfna deyakomu poryadkovomu chislu Dovedennya Redaguvati Nehaj S displaystyle S leq nbsp bude cilkom vporyadkovanoyu mnozhinoyu Mi pochnemo z vikoristannya transfinitnoyi rekursiyi dlya pobuduvannya poslidovnostej pidmnozhin S displaystyle S leq nbsp Baza Nehaj S 0 displaystyle S 0 emptyset nbsp Perehid do nastupnogo Dlya vsih poryadkovih chisel a displaystyle alpha nbsp viznachimo x a displaystyle x alpha nbsp yak najmenshij element z S S a displaystyle S setminus S alpha nbsp za umovi sho S S a displaystyle S setminus S alpha neq emptyset nbsp Todi S a 1 S a x a displaystyle S alpha 1 S alpha cup x alpha nbsp Perehid do granici Dlya vsih granichnih poryadkovih chisel l displaystyle lambda nbsp viznachiti S l a lt l S a displaystyle S lambda cup alpha lt lambda S alpha nbsp Cej proces maye zavershitis na pevnomu poryadkovomu chisli b displaystyle beta nbsp de S S b displaystyle S S beta emptyset nbsp Yaksho ni todi poryadkovi chisla mozhna vklasti v S displaystyle S nbsp z chogo viplivaye sho S displaystyle S nbsp zanadto velike shob buti mnozhinoyu sho protirichit umovi teoremi Teper pokazhemo sho dlya kozhnogo a lt b displaystyle alpha lt beta nbsp mnozhina S a displaystyle S alpha nbsp poryadkovo izomorfna a displaystyle alpha nbsp Baza S 0 displaystyle S 0 emptyset nbsp poryadkovo izomorfna 0 displaystyle 0 emptyset nbsp cherez porozhnyu fuknciyu yaka monotonna i yedina Perehid do nastupnogo Pripustimo sho S a displaystyle S alpha nbsp unikalno poryadkovo izomorfna pevnomu a displaystyle alpha nbsp dlya deyakogo poryadkovogo chisla a lt b displaystyle alpha lt beta nbsp Tobto f a S a a displaystyle f alpha S alpha rightarrow alpha nbsp ce unikalnij izomorfizm poryadku Teper za viznachennyam S a 1 S a x a displaystyle S alpha 1 S alpha cup x alpha nbsp de x a displaystyle x alpha nbsp ce najmenshij element S S a displaystyle S setminus S alpha nbsp Zvidsi vidno sho x a displaystyle x alpha nbsp ce najbilshij element z S a 1 displaystyle S alpha 1 nbsp Tomu mi viznachimo funkciyu f a 1 S a 1 a 1 displaystyle f alpha 1 S alpha 1 rightarrow alpha 1 nbsp yak f a 1 x f a x displaystyle f alpha 1 x f alpha x nbsp dlya vsih x S a displaystyle x in S alpha nbsp i f a 1 x a a displaystyle f alpha 1 x alpha alpha nbsp Ochevidno sho f a 1 displaystyle f alpha 1 nbsp ce poryadkovij izomorfizm i vin maye buti unikalnim bo maksimalni elementi vidpovidayut odin odnomu i f a displaystyle f alpha nbsp unikalna Perehid do granici Nehaj l b displaystyle lambda leq beta nbsp ce granichne poryadkove chislo i pripustimo sho f a S a a displaystyle f alpha S alpha rightarrow alpha nbsp ce unikalnij izomorfizm poryadku dlya kozhnogo a lt l displaystyle alpha lt lambda nbsp Zauvazhimo sho z togo sho S g S d displaystyle S gamma subset S delta nbsp pri g lt d displaystyle gamma lt delta nbsp viplivaye sho ci izomorfizmi poryadku mayut buti uzgodzhenimi tobto yaksho g lt d displaystyle gamma lt delta nbsp to f g 1 x g f d x g displaystyle f gamma 1 x gamma f delta x gamma nbsp Teper viznachimo f l S l l displaystyle f lambda S lambda rightarrow lambda nbsp yak f l x a f a 1 x a displaystyle f lambda x alpha f alpha 1 x alpha nbsp Nehaj f l x g f l x d displaystyle f lambda x gamma f lambda x delta nbsp dlya g d lt l displaystyle gamma leq delta lt lambda nbsp Za oznachennyam f l x g f g 1 x g displaystyle f lambda x gamma f gamma 1 x gamma nbsp i f l x d f d 1 x d displaystyle f lambda x delta f delta 1 x delta nbsp Ce daye rivnistf g 1 x g f d 1 x d displaystyle f gamma 1 x gamma f delta 1 x delta nbsp dd Zavdyaki tomu sho funkciyi uzgodzheni f g 1 x g f d 1 x g displaystyle f gamma 1 x gamma f delta 1 x gamma nbsp Todi f d 1 x g f d 1 x d displaystyle f delta 1 x gamma f delta 1 x delta nbsp otzhe x g x d displaystyle x gamma x delta nbsp i tomu f l displaystyle f lambda nbsp in yektivna Teper nehaj a l displaystyle alpha in lambda nbsp cherez te sho l displaystyle lambda nbsp skladayetsya z usih poryadkovih chisel menshih nizh l displaystyle lambda nbsp mayemo sho a lt l displaystyle alpha lt lambda nbsp A z togo sho l displaystyle lambda nbsp ce granichne poryadkove chislo viplivaye sho a 1 lt l displaystyle alpha 1 lt lambda nbsp Otzhe f l x a f a 1 x a a displaystyle f lambda x alpha f alpha 1 x alpha alpha nbsp i tomu f l displaystyle f lambda nbsp syur yektivna Nehaj x g lt x d S l displaystyle x gamma lt x delta in S lambda nbsp Todi f l x l f d 1 x g lt f d 1 x d displaystyle f lambda x lambda f delta 1 x gamma lt f delta 1 x delta nbsp otzhe f displaystyle f nbsp zberigaye poryadok Nasamkinec za tiyeyu zh argumentaciyeyu f l displaystyle f lambda nbsp unikalna bo f l displaystyle f lambda nbsp ce izomorfizm poryadku i jogo zvuzhennya do S a displaystyle S alpha nbsp musit buti unikalnim izomorfizmom poryadku z a displaystyle alpha nbsp Otzhe za dopomogoyu transfinitnoyi indukciyi mi doveli sho isnuye unikalnij izomorfizm poryadku z S b displaystyle S beta nbsp na b displaystyle beta nbsp Zavdyaki sposobu pobudovi ochevidno sho isnuye izomorfizm poryadku z S displaystyle S leq nbsp na b displaystyle beta nbsp a otzhe isnuye unikalnij izomorfizm poryadku z S displaystyle S leq nbsp na b displaystyle beta nbsp z chogo viplivaye sho S displaystyle S leq nbsp maye tip poryadku b displaystyle beta nbsp Posilannya RedaguvatiWeisstein Eric W Tip poryadku angl na sajti Wolfram MathWorld Otrimano z https uk wikipedia org w index php title Tip poryadku amp oldid 36798450
