On properties of the Hasse diagrams of NP-critical posets of order less than 8
DOI:
https://doi.org/10.17721/1812-5409.2024/1.5Keywords:
poset, critical and supercritical poset;, dyality, Hasse diagram, minimax equivalenceAbstract
M. Kleiner proved that a poset has finite representation type if and only if it does not contain subposets of the form (1, 1, 1, 1) =, (2, 2, 2) =, (1, 3, 3) =, (1, 2, 5) = and (И, 4) =, which are called the critical posets. A generalization of this criterion to the tame case was obtained by L. O. Nazarova. The corresponding posets are called supercritical and consist of the posets (1, 1, 1, 1, 1) =, (1, 1, 1, 2) =, (2, 2, 3) =, (1, 3, 4) =, (1, 2, 6) = and (И, 5) =. The critical and supercritical posets are also (respectively) critical respect to weakly positivity and weakly non-negativity of the Tits quadratic form which is given by the equality.
In the case of substitution on positivity and nonnegativity, such posets which are called P-critical and NP-critical, respectively, were described by the author together with V. M. Bondarenko (their number, up to isomorphism and duality, is 75 and 115). This article studies combinatorial properties of the Hasse diagrams of NP-critical posets.
Pages of the article in the issue: 33 - 35
Language of the article: English
References
Kleiner M. M. (1972) Partially ordered sets of finite type. Zap. Nauch. Semin. LOMI, 28, 32–41 [in Russian].
Bondarenko V. M., & Styopochkina M. V. (2005) (Min, max)-equivalence of partially ordered sets and the Tits quadratic form. Problems of Analysis and Algebra: Zb. Pr. Inst. Mat. NAN Ukr., 2, №3, 18-58 [in Russian].
Bondarenko V. M. (2005) On (min, max)-equivalence of posets and applications to the Tits forms. Bull. of Taras Shevchenko University of Kiev (series: Physics& Mathematics), 1, 24-25.
Nazarova L. A. (1975) Partially ordered sets of infinite type. Izv. Akad. Nauk SSSR Ser. Mat., 39, №5, 963-991 [in Russian].
Bondarenko V. M., & Styopochkina M. V. (2008) (Min, max)-equivalency of posets and nonnegative Tits forms. Ukrainian Math. J., 60,№9, 1349-1359. https://doi.org/10.1007/s11253-009-0147-7
Bondarenko V. M., & Styopochkina M. V. (2009) Description of posets critical with respect to the nonnegativity of the quadratic Tits form. Ukrainian Math. J., 61, №5, 734-746. https://doi.org/10.1007/s11253-009-0245-6
Bondarenko V. M., Chervyakov I. V., & Styopochkina M. V. (2015) On properties of the Hasse diagram of P-critical posets. Nauk. Visn. Uzhgorod. Univ., Ser. Mat. and Imfor, 26, №1, 12-15.
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