A semigroup of paths on a sequence of uniformly elliptic complexes

I. A. Ivanov-Pogodaev

Funkcionalʹnyj analiz i ego priloženiâ, Tome 57 (2023) no. 2, pp. 41-74.

Résumé

The work is devoted to solving a problem of L. N. Shevrin and M. V. Sapir (Question 3.81b of the Sverdlovsk Notebook), namely, to constructing a finitely presented infinite nil-semigroup satisfying the identity $x^9 = 0$. This problem is solved with the help of geometric methods of the theory of tilings and aperiodic tessellations. A semigroup of paths on a tiling, under certain conditions, inherits some properties of the tiling itself. Moreover, the defining relations in the semigroup correspond to a set of equivalent paths on the tiling. The relationship between the geometric and the automaton approaches previously used in the construction of finitely presented objects is discussed. As noted by S. P. Novikov, the property of determinacy in the coloring of partition nodes and its extension inward is very similar to properties of a solution of a partial differential equation with a given boundary condition. The author believes that understanding this relationship between the theories of aperiodic mosaics and their arrangements and the theory of numerical methods and grids is very promising.

Détail

Keywords: aperiodic tiling, determinacy, substitution complex, finitely presented semigroup, Burnside-type problem, nil-semigroup.

Bibliographie

[1] S. I. Adyan, “Problema Bernsaida i svyazannye s nei voprosy”, UMN, 65:5(395) (2010), 5–60 | DOI | MR | Zbl

[2] S. I. Adyan, “Novye otsenki nechetnykh periodov beskonechnykh bernsaidovykh grupp”, Trudy MIAN, 289 (2015), 41–82 | DOI | Zbl

[3] S. I. Adyan, Problema Bernsaida i tozhdestva v gruppakh, Nauka, M., 1975

[4] A. Ya. Belov, “Lineinye rekurrentnye uravneniya na dereve”, Matem. zametki, 78:5, 643–651 | DOI | MR | Zbl

[5] A. Ya. Belov, “Problemy bernsaidovskogo tipa, teoremy o vysote i o nezavisimosti”, Fundament. i prikl. matem., 13:5 (2007), 19–79

[6] A. Ya. Belov, M. I. Kharitonov, “Subeksponentsialnye otsenki v teoreme Shirshova o vysote”, Matem. sb., 203:4 (2012), 81–102 | DOI | Zbl

[7] A. Ya. Belov-Kanel, I. A. Ivanov-Pogodaev, “Konstruktsiya beskonechnoi konechno opredelennoi nilpolugruppy”, Dokl. RAN, ser. matem., inform., prots. upr., 101:2 (2020), 81–85 | MR | Zbl

[8] V. Ya. Belyaev, “Vlozhimost rekursivno opredelennykh inversnykh polugrupp v konechno opredelennye”, Sib. matem. zhurn., 25:2 (1984), 50–54 | MR | Zbl

[9] E. S. Golod, “O nil-algebrakh i finitno-approksimiruemykh $p$-gruppakh”, Izv. AN SSSR, ser. matem., 28:2 (1964), 273–276

[10] E. S. Golod, I. R. Shafarevich, “O bashne polei klassov”, Izv. AN SSSR, ser. matem., 28:2 (1964), 261–272 | Zbl

[11] Dnestrovskaya tetrad: Nereshennye problemy teorii kolets i modulei, izd. 4, In-t matematiki SO RAN, Novosibirsk, 1993

[12] I. A. Ivanov-Pogodaev, Mashina Minskogo, svoistva nilpotentnosti i razmernost Gelfanda–Kirillova v konechno opredelennykh polugruppakh, Diss. kand. fiz. mat. nauk, M., MGU, 2006

[13] I. A. Ivanov-Pogodaev, “O determinirovannosti putei na podstanovochnykh kompleksakh”, Dokl. RAN (to appear)

[14] I. A. Ivanov-Pogodaev, A. Ya. Kanel-Belov, “Konechno opredelennaya nilpolugruppa: kompleksy s ravnomernoi elliptichnostyu”, Izv. RAN, ser. matem., 85:6 (2021), 126–163 | DOI | MR | Zbl

[15] I. A. Ivanov-Pogodaev, A. Ya. Kanel-Belov, “Determinirovannaya raskraska semeistva kompleksov”, Fundament. prikl. matem. (to appear)

[16] N. K. Iyudu, “Algoritmicheskaya razreshimost problemy raspoznavaniya delitelei nulya v odnom klasse algebr”, Fundament. i prikl. matem., 1:2 (1995), 541–544 | MR | Zbl

[17] N. K. Iyudu, Standartnye bazisy i raspoznavaemost svoistv algebr, zadannykh kopredstavleniem, Diss. kand. fiz. mat. nauk, MGU, M., 1996

[18] A. I. Kostrikin, Vokrug Bernsaida, Nauka, M., 1986 | MR

[19] I. G. Lysenok, “Beskonechnost bernsaidovykh grupp perioda $2^k$ pri $k > 13$”, UMN, 47:2 (1992), 201–202 | MR | Zbl

[20] I. G. Lysenok, “Beskonechnye bernsaidovy gruppy chetnogo perioda”, Izv. RAN, ser. matem., 60:3 (1996), 3–224 | DOI | MR | Zbl

[21] P. S. Novikov, S. I. Adyan, “O beskonechnykh periodicheskikh gruppakh (I–III)”, Izv. AN SSSR, ser. matem., 32:1 (1968), 212–244 | MR

[22] A. Yu. Olshanskii, “O teoreme Novikova–Adyana”, Matem. sb., 118(160):2(6) (1982), 203–235 | MR

[23] A. Yu. Olshanskii, Geometriya opredelyayuschikh sootnoshenii v gruppakh, Nauka, M., 1989 | MR

[24] D. I. Piontkovskii, “Bazis Grëbnera i kogerentnost monomialnoi assotsiativnoi algebry”, Fundament. i prikl. matem., 2:2 (1996), 501–509 | MR | Zbl

[25] D. I. Piontkovskii, “Nekommutativnye bazisy Grebnera, kogerentnost assotsiativnykh algebr i delimost v polugruppakh”, Fundament. i prikl. matem., 7:2 (2001), 495–513 | MR | Zbl

[26] I. N. Sanov, “Reshenie problemy Bernsaida dlya pokazatelya $4$”, Uchen. zap. Leningr. un-ta, ser. matem., 10 (1940), 166–170 | MR | Zbl

[27] Sverdlovskaya tetrad: Nereshennye zadachi teorii polugrupp, no. 3, Izd-vo Uralsk. gos. un-ta, Sverdlovsk, 1989

[28] V. A. Ufnarovskii, “Kombinatornye i asimptoticheskie metody v algebre”, Itogi nauki i tekhn. Sovrem. probl. matem. Fundam. napravleniya, no. 57, VINITI, M., 1990, 5–177 | MR

[29] V. A. Ufnarovskii, “O roste algebr”, Vestnik MGU. ser. 1, 1978, no. 4, 59–65 | MR | Zbl

[30] A. I. Shirshov, “O nekotorykh neassotsiativnykh nil-koltsakh i algebraicheskikh algebrakh”, Matem. sb., 41:3 (1957), 381–394 | Zbl

[31] A. I. Shirshov, “O koltsakh s tozhdestvennymi sootnosheniyami”, Matem. sb., 43:2 (1957), 277–283 | Zbl

[32] V. V. Schigolev, “O nil i nilpotentnykh konechnoopredelennykh algebrakh”, Fundament. i prikl. matem., 6:4 (2000), 1239–1245 | MR

[33] A. Kanel-Belov, H. L. Rowen, “Perspectives on Shirshov's height theorem”, Selected Works of A. I. Shirshov, Birkhäuser Verlag, 2009, 3–20 | MR

[34] A. Ya. Belov, V. V. Borisenko, V. N. Latysev, “Monomial algebras”, J. Math. Sci. (NY), 87:3 (1997), 3463–3575 | DOI | MR | Zbl

[35] A. Ya. Belov, I. A. Ivanov, “Construction of semigroups with some exotic properties”, Comm. Algebra, 31:2 (2003), 673–696 | DOI | MR | Zbl

[36] A. Ya. Belov, I. A. Ivanov, “Construction of semigroups with some exotic properties”, Acta Appl. Math., 85:1–3 (2005), 49–56 | DOI | MR | Zbl

[37] L. A. Bokut, G. P. Kukin, Algoritmic and Combinatorial Aldebra, Math. and Its Applications, no. 255, Kluwer Acad. Publishers Group, Dordrecht, 1994 | MR

[38] S. V. Ivanov, “The free Burnside groups of sufficiently large exponents”, Internat. J. Algebra Comput., 4:1–2 (1994), 1–308 | DOI | MR | Zbl

[39] S. V. Ivanov, “On the Burnside problem on periodic groups”, Bull. Amer. Math. Soc. (N.S.), 27:2 (1992), 257–260 | DOI | MR | Zbl

[40] I. Ivanov-Pogodaev, S. Malev, “Finite Groebner basis algebra with unsolvable problems of nilpotency and zero divisors”, J. Algebra, 508 (2018), 575–588 | DOI | MR | Zbl

[41] I. Ivanov-Pogodaev, S. Malev, O. Sapir, “A construction of a finitely presented semigroup containing an infinite square-free ideal with zero multiplication”, Internat. J. Algebra Comput., 28:8 (2018), 1565–1573 | DOI | MR | Zbl

[42] M. Jr. Hall, “Solution of the Burnside problem for exponent $6$”, Proc. Nat. Acad. Sci. USA, 43 (1957), 751–753 | DOI | MR | Zbl

[43] A. R. Kemer, “Comments on the Shirshov's height theorem”, Selected works of A. I. Shirshov, Birkhüser Verlag, 2009, 41–48 | MR

[44] G. P. Kukin, “The variety of all rings has Higman's property”, Third Siberian School Algebra and Analysis, Irkutsk State Univ., Irkutsk, 1989, 91–101 | MR

[45] O. G. Kharlampovich, M. V. Sapir, “Algorithmic problems in varieties”, Internat. J. Algebra Comput., 5:4–5 (1995), 379–602 | DOI | MR | Zbl

[46] A. Yu. Ol'shanskii, M. V. Sapir, “Non-amenable finitely presented torsion-by-cyclic groups”, Publ. Math. Inst. Hautes Étud. Sci., 96:1 (2003), 43–169 | DOI | MR

[47] M. V. Sapir, “Algorithmic problems for amalgams of finite semigroups”, J. Algebra, 229:2 (2000), 514–531 | DOI | MR | Zbl

[48] M. V. Sapir, Combinatorial Algebra: Syntax and Semantics, with contributions by Victor S. Guba and Mikhail V. Volkov, Springer, Cham, 2014 | MR | Zbl

[49] E. I. Zelmanov, “On the nilpotency of nil algebras”, Lect. Notes of Math., 1352, Springer-Verlag, Berlin, 1988, 227–240 | DOI | MR