Dehornoy tartibi - Dehornoy order

In matematik maydoni ortiqcha oro bermay nazariyasi, Dehornoy tartibi chap o'zgarmasdir umumiy buyurtma ustida to'quv guruhi, tomonidan topilgan Patrik Dehornoy.^[1]^[2] Dehornoy tomonidan ishlatilgan braid guruhidagi tartibni asl kashfiyoti ulkan kardinallar, ammo hozirda uning yana bir necha oddiy konstruktsiyalari mavjud.^[3]

Ta'rif

Aytaylik ${ displaystyle sigma _ {1}, nuqtalar, sigma _ {n-1}}$ braid guruhining odatiy generatorlari ${ displaystyle B_ {n}}$ kuni ${ displaystyle n}$ torlar. A ni aniqlang ${ displaystyle sigma _ {i}}$ - ijobiy so'z elementlarda kamida bitta iborani tan oladigan ortiqcha oro bermay bo'lish ${ displaystyle sigma _ {1}, nuqtalar, sigma _ {n-1}}$ va ularning teskari tomonlari, masalan, so'z o'z ichiga oladi ${ displaystyle sigma _ {i}}$ , lekin o'z ichiga olmaydi ${ displaystyle sigma _ {i} ^ {- 1}}$ na ${ displaystyle sigma _ {j} ^ { pm 1}}$ uchun ${ displaystyle j$ .

To'plam ${ displaystyle P}$ Dehornoy tartibidagi ijobiy elementlarning a sifatida yozilishi mumkin bo'lgan elementlar sifatida aniqlanadi ${ displaystyle sigma _ {i}}$ - ba'zilar uchun ijobiy so'z ${ displaystyle i}$ .

To'plam ${ displaystyle P}$ qondiradi ${ displaystyle PP subseteq P}$ , to'plamlar ${ displaystyle P}$ , ${ displaystyle {1 }}$ va ${ displaystyle P ^ {- 1}}$ disjoint ("acyclicity property") va to'quv guruhi ${ displaystyle P}$ , ${ displaystyle {1 }}$ va ${ displaystyle P ^ {- 1}}$ ("taqqoslash xususiyati"). Bu xususiyatlar shuni anglatadiki, agar biz " ${ displaystyle a$ " anglatmoq " ${ displaystyle a ^ {- 1} b in P}$ "keyin biz to'qilgan guruh bo'yicha chap-o'zgarmas umumiy buyurtmani olamiz. Masalan, ${ displaystyle sigma _ {1} < sigma _ {2} sigma _ {1}}$ chunki sochlar so'zi ${ displaystyle sigma _ {1} ^ {- 1} sigma _ {2} sigma _ {1}}$ emas ${ displaystyle sigma _ {1}}$ -pozitiv, lekin, braid munosabatlari bilan, ga teng ${ displaystyle sigma _ {1}}$ - ijobiy so'z ${ displaystyle sigma _ {2} sigma _ {1} sigma _ {2} ^ {- 1}}$ yotadi ${ displaystyle P}$ .

Tarix

To'siq nazariyasi kabi turli xil "giper-cheksizlik" tushunchalarining faraziy mavjudligini taqdim etadi katta kardinallar. 1989 yilda ana shunday tushunchalardan biri aksioma ekanligi isbotlandi ${ displaystyle I_ {3}}$ , asiklik ssilka deb ataladigan algebraik strukturaning mavjudligini anglatadi, bu esa o'z navbatida aniqlik ning so'z muammosi chap o'z-o'zini tarqatish qonuni uchun ${ displaystyle LD: x (yz) = (xy) (xz)}$ , katta kardinallar bilan bog'liq bo'lmagan apriori bo'lgan mulk.^[4]^[5]

1992 yilda Dehornoy ma'lum bir mahsulotni taqdim etish orqali asiklik ssilkaning namunasini ishlab chiqardi guruxsimon ${ displaystyle { mathcal {G}} _ {LD}}$ ning geometrik tomonlarini aks ettiruvchi ${ displaystyle LD}$ qonun. Natijada, asiklik javon qurildi to'quv guruhi ${ displaystyle B _ { infty}}$ , bu sodir bo'ladi ${ displaystyle { mathcal {G}} _ {LD}}$ , va bu to'g'ridan-to'g'ri to'qish tartibining mavjudligini anglatadi.^[2] To'qimachilik tartibi katta kardinal taxmin tugatilgandan so'ng paydo bo'lganligi sababli, ortiqcha oro bermay tartibi va asiklik raft o'rtasidagi bog'liqlik faqat o'rnatilgan nazariyadan kelib chiqadigan dastlabki muammo orqali aniq bo'ldi.^[6]

Xususiyatlari

Buyurtmaning mavjudligi shuni ko'rsatadiki, har bir braid guruhi ${ displaystyle B_ {n}}$ tartibli guruh va shu sababli algebralar ${ displaystyle mathbb {Z} B_ {n}}$ va ${ displaystyle mathbb {C} B_ {n}}$ nol bo'luvchisi yo'q.
Uchun ${ displaystyle n geqslant 3}$ , Dehornoy buyrug'i o'ng tomonda o'zgarmas emas: bizda bor ${ displaystyle sigma _ {2} < sigma _ {1}}$ va ${ displaystyle sigma _ {2} sigma _ {1}> sigma _ {1} ^ {2}}$ . Yaxshiyamki, buyruq yo'q ${ displaystyle B_ {n}}$ bilan ${ displaystyle n geqslant 3}$ ikkala tomon ham o'zgarmas bo'lishi mumkin.
Uchun ${ displaystyle n geqslant 3}$ , Dehornoy buyrug'i na Arximed, na Konradiydir: braidlar mavjud ${ displaystyle beta _ {1}, beta _ {2}}$ qoniqarli ${ displaystyle beta _ {1} ^ {p} < beta _ {2}}$ har bir kishi uchun ${ displaystyle p}$ (masalan; misol uchun, ${ displaystyle beta _ {1} = sigma _ {2}}$ va ${ displaystyle beta _ {2} = sigma _ {1}}$ ) va braidlar ${ displaystyle beta _ {1}, beta _ {2}}$ dan katta ${ displaystyle 1}$ qoniqarli ${ displaystyle beta _ {1}> beta _ {2} beta _ {1} ^ {p}}$ har bir kishi uchun ${ displaystyle p}$ (masalan; misol uchun, ${ displaystyle beta _ {1} = sigma _ {2} ^ {- 1} sigma _ {1}}$ va ${ displaystyle beta _ {2} = sigma _ {2} ^ {- 2} sigma _ {1}}$ ).
Dehornoy buyrug'i - bu ijobiy ortiqcha oro bermay monoid bilan cheklangan holda yaxshi tartib ${ displaystyle B_ {n} ^ {+}}$ tomonidan yaratilgan ${ displaystyle sigma _ {1}, ldots, sigma _ {n-1}}$ (Richard Laver ^[7]). Dehornoy buyurtmasining buyurtma turi cheklangan ${ displaystyle B_ {n} ^ {+}}$ tartibli hisoblanadi ${ displaystyle omega ^ { omega ^ {n-2}}}$ (Serj Burkel.) ^[8]).
Dehornoy buyrug'i ikkitomonlama ijobiy braid monoid bilan cheklangan bo'lsa ham yaxshi buyurtma hisoblanadi ${ displaystyle B_ {n} ^ {* +}}$ elementlari tomonidan hosil qilingan ${ displaystyle sigma _ {i} dots sigma _ {j-1} sigma _ {j} sigma _ {j-1} ^ {- 1} dots sigma _ {i} ^ {- 1 }}$ bilan ${ displaystyle 1 leqslant i$ , va Dehornoy buyurtmasining buyurtma turi cheklangan ${ displaystyle B_ {n} ^ {* +}}$ ham ${ displaystyle omega ^ { omega ^ {n-2}}}$ (Jan Fromentin ^[9]).
Ikkilik munosabat sifatida Dehornoy buyrug'i hal qilinadi. Qarorning eng yaxshi algoritmi Dynnikovning tropik formulalariga asoslangan (Ivan Dynnikov,^[10] ning XII bobiga qarang ^[3]); hosil bo'lgan algoritm bir xil murakkablikni tan oladi ${ displaystyle O ( ell ^ {2})}$ .

Tugun nazariyasi bilan bog'lanish

Ruxsat bering ${ displaystyle Delta _ {n}}$ Garside-ning asosiy yarim burilish sochlari bo'ling. Har qanday ortiqcha oro bermay ${ displaystyle beta}$ noyob intervalda yotadi ${ displaystyle [ Delta _ {n} ^ {2m}, Delta _ {n} ^ {2m + 2})}$ ; butun sonni chaqiring ${ displaystyle m}$ The Dehornoy qavat ning ${ displaystyle beta}$ , belgilangan ${ displaystyle lfloor beta rfloor}$ . Keyinchalik katta qavatdagi braidlarning bog'lanishini yopish yaxshi, ya'ni xususiyatlari ${ displaystyle { widehat { beta}}}$ dan osongina o'qish mumkin ${ displaystyle beta}$ . Mana ba'zi misollar.
Agar ${ displaystyle vert lfloor beta rfloor vert> 1}$ ushlab turadi, keyin ${ displaystyle { widehat { beta}}}$ oddiy, oddiy va ahamiyatsiz (Andrey Malyutin va Nikita Netstetaev) ^[11]).
Agar ${ displaystyle vert lfloor beta rfloor vert> 1}$ ushlab turadi va ${ displaystyle { widehat { beta}}}$ tugun, keyin ${ displaystyle { widehat { beta}}}$ torik tugun va agar shunday bo'lsa ${ displaystyle beta}$ davriy, ${ displaystyle { widehat { beta}}}$ a sun'iy yo'ldosh tuguni agar va faqat agar ${ displaystyle beta}$ kamaytirilishi mumkin va ${ displaystyle { widehat { beta}}}$ agar va faqat shunday bo'lsa, hiperbolikdir ${ displaystyle beta}$ soxta Anosov (Tetsuya Ito) ^[12]).

Adabiyotlar

^ Dehornoy, Patrik (1992), "Deux propriétés des groupes de tresses", Comptes Rendus de l'Académie des Sciences, Série I, 315 (6): 633–638, ISSN 0764-4442, JANOB 1183793
^ ^a ^b Dehornoy, Patrik (1994), "Braid guruhlari va chap tarqatish operatsiyalari", Amerika Matematik Jamiyatining operatsiyalari, 345 (1): 115–150, doi:10.2307/2154598, JSTOR 2154598, JANOB 1214782
^ ^a ^b Dehornoy, Patrik; Dynnikov, Ivan; Rolfsen, Deyl; Wiest, Bert (2008), Braidlarni buyurtma qilish, Matematik tadqiqotlar va monografiyalar, 148, Providence, R.I .: Amerika matematik jamiyati, ISBN 978-0-8218-4431-1, JANOB 2463428
^ Dehornoy, Patrik (1989), "Sur la structure des gerbes libres", Comptes Rendus de l'Académie des Sciences, Série I, 309 (3): 143–148, JANOB 1005627
^ Laver, Richard (1992), "Chap tarqatish qonuni va elementar birikmalar algebrasining erkinligi", Matematikaning yutuqlari, 91 (2): 209–231, doi:10.1016 / 0001-8708 (92) 90016-E, hdl:10338.dmlcz / 127389, JANOB 1149623
^ Dehornoy, Patrik (1996), "To'plamlar nazariyasidan yana bir foydalanish", Ramziy mantiq byulleteni, 2 (4): 379–391, doi:10.2307/421170, JSTOR 421170, JANOB 1321290
^ Laver, Richard (1996), "chap taqsimlovchi tuzilmalardagi to'qish guruhi harakatlari va to'qish guruhlarida quduq buyurtmalari", Sof va amaliy algebra jurnali, 108: 81–98, doi:10.1016/0022-4049(95)00147-6, JANOB 1382244
^ Burckel, Serj (1997), "Ijobiy braidlar bo'yicha buyurtma", Sof va amaliy algebra jurnali, 120 (1): 1–17, doi:10.1016 / S0022-4049 (96) 00072-2, JANOB 1466094
^ Fromentin, Jan (2011), "Har bir braid qisqa sigma aniq ifodasini tan oladi", Evropa matematik jamiyati jurnali, 13 (6): 1591–1631, doi:10.4171 / JEMS / 289 | mr = 2835325
^ Dynnikov, Ivan (2002), "Yang-Baxter xaritasi va Dehornoy buyurtmasi to'g'risida", Rossiya matematik tadqiqotlari, 57 (3): 151–152, doi:10.1070 / RM2002v057n03ABEH000519, JANOB 1918864
^ Malyutin, Andrey; Netsvetaev, Nikita Yu. (2003), "To'qimachilik guruhidagi dehornoy tartib va yopiq sochlarning o'zgarishi", Rossiĭskaya Akademiya Nauk. Algebra i Analiz, 15 (3): 170–187, doi:10.1090 / S1061-0022-04-00816-7, JANOB 2052167
^ Ito, Tetsuya (2011), "To'siqqa buyurtma berish va tugunlar", Tugunlar nazariyasi jurnali va uning samaralari, 20 (9): 1311–1323, arXiv:0805.2042, doi:10.1142 / S0218216511009169, JANOB 2844810

Qo'shimcha o'qish

Kassel, Kristian (2002), "L'ordre de Dehornoy sur les tresses", Asterisk (276): 7–28, ISSN 0303-1179, JANOB 1886754
Dehornoy, Patrik (1997), "Braidlarni taqqoslashning tezkor usuli", Matematikaning yutuqlari, 125 (2): 200–235, doi:10.1006 / aima.1997.1605, JANOB 1434111

[1] Dehornoy, Patrik (1992), "Deux propriétés des groupes de tresses", Comptes Rendus de l'Académie des Sciences, Série I, 315 (6): 633–638, ISSN 0764-4442, JANOB 1183793

[Dehornoy94-2] Dehornoy, Patrik (1994), "Braid guruhlari va chap tarqatish operatsiyalari", Amerika Matematik Jamiyatining operatsiyalari, 345 (1): 115–150, doi:10.2307/2154598, JSTOR 2154598, JANOB 1214782

[DDRW08-3] Dehornoy, Patrik; Dynnikov, Ivan; Rolfsen, Deyl; Wiest, Bert (2008), Braidlarni buyurtma qilish, Matematik tadqiqotlar va monografiyalar, 148, Providence, R.I .: Amerika matematik jamiyati, ISBN 978-0-8218-4431-1, JANOB 2463428

[4] Dehornoy, Patrik (1989), "Sur la structure des gerbes libres", Comptes Rendus de l'Académie des Sciences, Série I, 309 (3): 143–148, JANOB 1005627

[5] Laver, Richard (1992), "Chap tarqatish qonuni va elementar birikmalar algebrasining erkinligi", Matematikaning yutuqlari, 91 (2): 209–231, doi:10.1016 / 0001-8708 (92) 90016-E, hdl:10338.dmlcz / 127389, JANOB 1149623

[6] Dehornoy, Patrik (1996), "To'plamlar nazariyasidan yana bir foydalanish", Ramziy mantiq byulleteni, 2 (4): 379–391, doi:10.2307/421170, JSTOR 421170, JANOB 1321290

[7] Laver, Richard (1996), "chap taqsimlovchi tuzilmalardagi to'qish guruhi harakatlari va to'qish guruhlarida quduq buyurtmalari", Sof va amaliy algebra jurnali, 108: 81–98, doi:10.1016/0022-4049(95)00147-6, JANOB 1382244

[8] Burckel, Serj (1997), "Ijobiy braidlar bo'yicha buyurtma", Sof va amaliy algebra jurnali, 120 (1): 1–17, doi:10.1016 / S0022-4049 (96) 00072-2, JANOB 1466094

[9] Fromentin, Jan (2011), "Har bir braid qisqa sigma aniq ifodasini tan oladi", Evropa matematik jamiyati jurnali, 13 (6): 1591–1631, doi:10.4171 / JEMS / 289 | mr = 2835325

[10] Dynnikov, Ivan (2002), "Yang-Baxter xaritasi va Dehornoy buyurtmasi to'g'risida", Rossiya matematik tadqiqotlari, 57 (3): 151–152, doi:10.1070 / RM2002v057n03ABEH000519, JANOB 1918864

[11] Malyutin, Andrey; Netsvetaev, Nikita Yu. (2003), "To'qimachilik guruhidagi dehornoy tartib va yopiq sochlarning o'zgarishi", Rossiĭskaya Akademiya Nauk. Algebra i Analiz, 15 (3): 170–187, doi:10.1090 / S1061-0022-04-00816-7, JANOB 2052167

[12] Ito, Tetsuya (2011), "To'siqqa buyurtma berish va tugunlar", Tugunlar nazariyasi jurnali va uning samaralari, 20 (9): 1311–1323, arXiv:0805.2042, doi:10.1142 / S0218216511009169, JANOB 2844810

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