On transitivity coefficients for minimal posets with non-positive quadratic Tits form
Анотація
Досліджуються комбінаторні властивості скінченних частково впорядкованих множин, пов’язаних з додатністю їх квадратичної форми Тітса (яка відіграє важливу роль у теорії матричних зображень частково впорядкованих множин). Для всіх мінімальних частково впорядкованих множин з недодатною квадратичною формою Тітса (такі множини називають P-критичними, і їх кількість, з точністю до ізоморфізму і дуальності, дорівнює 75) обчислено коефіцієнти транзитивності і встановлено деякі зв’язки між цими коефіцієнтами та висотами частково впорядкованих множин.
Зразок для цитування: V. M. Bondarenko, M. V. Styopochkina, “On transitivity coefficients for minimal posets with non-positive quadratic Tits form,” Мат. методи та фіз.-мех. поля, 64, No. 1, 5–14 (2021), https://doi.org/10.15407/mmpmf2021.64.1.5-14
Reprinted as: V. M. Bondarenko, M. V. Styopochkina, “On the transitivity coefficients for minimal posets with nonpositive quadratic Tits form,” J. Math. Sci., 274, No. 5, 583–593 (2023), https://doi.org/10.1007/s10958-023-06624-6
Ключові слова
Посилання
V. M. Bondarenko, A. G. Zavadskiĭ, L. A. Nazarova, “On representations of tame partially ordered sets”, Predstavl. Kvadrat. Formy, 75–106 (1979); English translation: Amer. Math. Soc. Transl. Ser. 2, 128, 55-78 (1986).
V. M. Bondarenko, M. V. Stepochkina, “(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, "On serial posets with positive-definite quadratic Tits form," Nelin. Kolyv., 9, No. 3, 320-325 (2006); English translation: Nonlinear Oscil., 9, No. 3, 312-316 (2006), https://doi.org/10.1007/s11072-006-0045-y
Yu. A. Drozd, “Coxeter transformations and representations of partially ordered sets,” Funkts. Anal. Prilozhen., 8, No. 3, 34–42 (1974); English translation: Funct. Anal. Appl., 8, No. 3, 219–225 (1974), https://doi.org/10.1007/BF01075695
L. A. Nazarova, A. V. Roiter, "Representations of partially ordered sets," Zap. Nauch. Semin. LOMI, 28, 5-31 (1972); English translation: J. Sov. Math., 3, No. 5, 585–606 (1975), https://doi.org/10.1007/BF01084662
M. Afkhami, K. Khashyarmanesh, F. Shahsavar, “On the intersection graphs associeted to posets,” Discuss. Math. Gen. Algebra Appl., 40, No. 1, 105–117 (2020), https://doi.org/10.7151/dmgaa.1322
V. M. Bondarenko, M. V. Styopochkina, "Coefficients of transitiveness of P-critical posets," in: Analysis and application: Zb. Pr. Inst. Mat. NAN Ukr., 14, No. 1, 46–51 (2017).
V. M. Bondarenko, M. V. Styopochkina, "On finite posets of inj-finite type and their Tits forms," Algebra Discrete Math., 5, No. 2, 17–21 (2006).
V. M. Bondarenko, M. V. Styopochkina, "On posets of width two with positive Tits form," Algebra Discrete Math., 4, No. 2, 20–35 (2005).
K. Bongartz, “Algebras and quadratic forms,” J. London Math. Soc., s2-28, No. 3, 461–469 (1983), https://doi.org/10.1112/jlms/s2-28.3.461
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
J. Cooper, P. Gartlan, H. Whitlatch, “A new characterization of v-posets,” Order, 37, No. 2, 371–387 (2020), https://doi.org/10.1007/s11083-019-09510-6
J. A. De la Pena, A. Skowroński, “The Tits and Euler forms of a tame algebra,” Math. Ann., 1999, 315, No. 1, 37–59, https://doi.org/10.1007/s002080050317
P. Draxler, J. A. De la Pena, “Tree algebras with non-negative Tits form,” Comm. Algebra, 28, No. 8, 3993–4012 (2000), https://doi.org/10.1080/00927870008827071
P. Gabriel, “Unzerlegbare Darstellungen I,” Manuscripta Math., 6, 71–103 (1972), https://doi.org/10.1007/BF01298413
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).
J. Kosakowska, D. Simson, “On Tits form and prinjective representations of posets of finite prinjective type,” Comm. Algebra, 26, No. 5, 1613–1623 (1998), https://doi.org/10.1080/00927879808826225
N. Kravitz, A. Sah, “Linear extension numbers of n-element posets,” Order, 38, No. 1, 49–66 (2021), https://doi.org/10.1007/s11083-020-09527-2
N. Marmaridis, “One point extensions of trees and quadratics forms,” Fund. Math., 134, No. 1, 15–35 (1990), https://doi.org/10.4064/fm-134-1-15-35
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
D. Simson, “A reduction functor, tameness, and Tits form for a class of orders,” J. Algebra, 174, No. 2, 430–452 (1995), https://doi.org/10.1006/jabr.1995.1133
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