Classification of the posets of minmax types which are symmetric oversupercritical posets of the eighth order

V. M. Bondarenko, M. V. Styopochkina

Анотація


Класифікація частково впорядкованих множин, мінімаксним типом яких є симетричні надсуперкритичні частково впорядковані множини порядку 8

 

Наведено класифікацію частково впорядкованих множин, що тісно пов’язані (сто­совно своїх квадратичних форм Тітса) з узагальненнями критичних і суперкритичних частково впорядкованих множин, які вперше появилися у критеріях Клейнера та Назарової стосовно зображувальних типів частково впорядкованих множин. Ці критерії були першими в новій теорії зображень, започаткованій Л. О. Назаровою та А. В. Ройтером у 1972 р. Метод мінімаксного ізоморфізму (запроваджений першим автором) відіграє основну роль у поданому дослідженні.

 

Зразок для цитування: V. M. Bondarenko, M. V. Styopochkina, “Classification of the posets of minmax types which are symmetric oversupercritical posets of the eighth order,” Мат. методи та фіз.-мех. поля, 66, No. 1-2, 5–15 (2023), https://doi.org/


Ключові слова


діаграма Гассе, квадратична форма Тітса, критичні, надкритичні та супернадкритичні множини, мінімаксний тип, коефіцієнт транзитивності, вузловий елемент, щільна підмножина

Посилання


V. V. Bondarenko, V. M. Bondarenko, M. V. Styopochkina, I. V. Chervyakov, "1-over-supercritical partially ordered sets with trivial group of automorphisms and min-equivalence. I," Nauk. Visn. Uzhgorod. Univ., Ser. Mat. Inf., 22, No. 2, 17–25 (2011) (in Russian).

V. M. Bondarenko, M. V. Styopochkina, “(Min, max)-equivalence of posets and quadratic Tits form,” in: Problems of Analysis and Algebra [in Russian], 2, No. 3, Inst. Mat. NAN Ukr., Kyiv (2005), pp. 18–58.

V. M. Bondarenko, M. V. Stepochkina, "(Min, max)-equivalence of posets and non-negative Tits forms," Ukr. Mat. Zh., 60, No. 9, 1157–1167 (2008) (in Russian); English translation: Ukr. Math. J., 60, No. 9., 1349–1359 (2008), https://doi.org/10.1007/s11253-009-0147-7

V. M. Bondarenko, M. V. Stepochkina, "Description of posets critical with respect to the nonnegativity of the quadratic Tits form," Ukr. Mat. Zh., 61, No. 5, P. 611-624 (2009) (in Russian); English translation: Ukr. Math. J., 61, No. 5, 734-746 (2009), https://doi.org/10.1007/s11253-009-0245-6

V. M. Bondarenko, M. V. Stoika, M. V. Styopochkina, “On combinatorial properties of the posets of oversupercritical MM-type of smallest order,” Nauk. Visn. Uzhhorod Univ. Ser. Mat. Inf., 42, No. 1, 7–11 (2023) (in Ukrainian), https://doi.org/10.24144/2616-7700.2023.42(1).7-11

M. M. Kleiner, “Partially ordered sets of finite type,” Zap. Nauch. Semin. Leningrad. Otdel. Mat. Inst. Steklov, 28, 32–41 (1972) (in Russian); English translation: J. Sov. Math., 3, No. 5, 607–615 (1975), https://doi.org/10.1007/BF01084663

L. A. Nazarova, “Partially ordered sets of infinite type,” Izv. Akad. Nauk SSSR. Ser. Mat., 39, No. 5, 963–991 (1975) (in Russian); English translation: Math.USSR-Izv., 9, No. 5, 911–938 (1975), https://doi.org/10.1070/IM1975v009n05ABEH001511

L. A. Nazarova, A. V. Roiter, “Representations of partially ordered sets,” Zap. Nauch. Semin. LOMI, 28, 5–31 (1972) (in Russian); English translation: J. Sov. Math., 3, No. 5, 585–606 (1975), https://doi.org/10.1007/BF01084662

V. M. Bondarenko, "On (min, max)-equivalence of posets and applications to the Tits forms," Visn. Kyiv Univ., Ser. Fiz. Mat., No. 1, 24-25 (2005).

V. M. Bondarenko, M. V. Styopochkina, “Coefficients of transitiveness of P-critical posets,” in: Analysis and application: Zb. Pr. Inst. Mat Nats. Akad. Nauk Ukr., 14, No. 1, 46–51 (2017).

V. M. Bondarenko, M. V. Styopochkina, “Combinatorial properties of P-posets of width 2,” Prykl. Probl. Mekh. Mat., Issue 15, 21–23 (2017).

V. M. Bondarenko, M. V. Styopochkina, “On classifying the non-Tits P-critical posets,” Algebra Discrete Math., 32, No. 2, 185–196 (2021), https://doi.org/10.12958/adm1912

V. M. Bondarenko, M. V. Styopochkina, “On posets of sixth order having oversupercritical MM-type,” Nauk. Visn. Uzhhorod Univ. Ser. Mat. Inf., 38, No. 1, 7–15 (2021), https://doi.org/10.24144/2616-7700.2021.38(1).7-15

V. M. Bondarenko, M. V. Styopochkina, "On properties of posets of MM-type (1, 2, 7)," Prykl. Probl. Mekh. Mat., Issue 17, 7–10 (2019), https://doi.org/10.15407/apmm2019.17.7-10

V. M. Bondarenko, M. V. Styopochkina, “On properties of posets of MM-type (1, 3, 5),” Nauk. Visn. Uzhhorod Univ. Ser. Mat. Inf., 32, No. 1, 50–53 (2018)

V. M. Bondarenko, M. V. Styopochkina, “On transitivity coefficient of posets of MM-type to be oversupercritical non-primitive,” Nauk. Visn. Uzhhorod Univ. Ser. Mat. Inf., 39, No. 2, 22–29 (2021), https://doi.org/10.24144/2616-7700.2021.39(2).22-29.

V. M. Bondarenko, M. V. Styopochkina, “On transitivity coefficients for minimal posets with non-positive quadratic Tits form,” Mat. Met. Fiz.-Mekh. Polya, 64, No. 1, 5-14, https://doi.org/10.15407/mmpmf2021.64.1.5-14. (2021); English translation: J. Math. Sci., 274, No. 5, 583-593 (2023), https://doi.org/10.1007/s10958-023-06624-6

V. M. Bondarenko, M. V. Styopochkina, “Strengthening of a theorem on Coxeter–Euclidean type of principal partially ordered sets,” Visn. Kyiv. Univ. im. Shevchenka, Ser. Fiz.-Mat. Nauky, No. 4, 8–15 (2018), https://doi.org/10.17721/1812-5409.2018/4.1

S. H. A. Borujeni, N. Bowler, “Investigating posets via their maximal chains,” Order, 37, No. 2, 299–309 (2020), https://doi.org/10.1007/s11083-019-09506-2

I. Chajda, D. Fazio, H. Länger, A. Ledda, J. Paseka, “Algebraic properties of para-orthomodular posets,” Log. J. IGPL, 30, No. 5, 840–869 (2022), https://doi.org/10.1093/jigpal/jzab024

K. L. Collins, A. N. Trenk, “The distinguishing number and distinguishing chromatic number for posets,” Order, 39, No. 3, 361–380 (2022), https://doi.org/10.1007/s11083-021-09583-2

J. Cooper, P. Gartland, H. Whitlatch, “A new characterization of -posets,” Order, 37, No. 2, 371–387 (2020), https://doi.org/10.1007/s11083-019-09510-6

V. I. Danilov, “Choice functions on posets,” Order, 40, No. 2, 387–396 (2023), https://doi.org/10.1007/s11083-022-09618-2

M. Gąsiorek, D. Simson, “One-peak posets with positive quadratic Tits form, their mesh translation quivers of roots, and programming in Maple and Python,” Linear Algebra Appl., 436, No. 7, 2240–2272 (2012), https://doi.org/10.1016/j.laa.2011.10.045

G. Guśpiel, P. Micek, A. Polak, “On an extremal problem for poset dimension,” Order, 35, No. 3, 489–493 (2018), https://doi.org/10.1007/s11083-017-9444-1

D. M. Howard, N. Streib, W. T. Trotter, B. Walczak, R. Wang, “Dimension of posets with planar cover graphs excluding two long incomparable chains,” J. Comb. Theory. Ser. A, 164, 1–23 (2019), https://doi.org/10.1016/j.jcta.2018.11.016

M. Kaniecki, J. Kosakowska, P. Malicki, G. Marczak, “A horizontal mesh algorithm for posets with positive Tits form,” Algebra Discrete Math., 22, No. 2, 240–261 (2016).

N. Kravitz, A. Sah, “Linear extension numbers of -element posets,” Order, 38, No. 1, 49–66 (2021), https://doi.org/10.1007/s11083-020-09527-2

A. Polak, D. Simson, “Coxeter spectral classification of almost TP-critical one-peak posets using symbolic and numeric computations,” Linear Algebra Appl., 445, 223–255 (2014), https://doi.org/10.1016/j.laa.2013.12.018

B. E. Tenner, “Interval posets of permutations,” Order, 39, No. 3, 523–536 (2022), https://doi.org/10.1007/s11083-021-09576-1


Посилання

  • Поки немає зовнішніх посилань.


Creative Commons License
Ця робота ліцензована Creative Commons Attribution 3.0 License.