Classification of the almost positive posets

• Date: 2025
• Main Authors: Bondarenko, Vitaliy M., Styopochkina, Maryna V.
• Format: Article
• Language: English
• Published: Lugansk National Taras Shevchenko University 2025
• Subjects: almost positive poset; principal poset; \(P\)-critical poset; minimax equivalence; Tits quadratic form; minimax \(d\)-system of generators; 06A07; 11E04
• Online Access: https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2391
• Journal Title: Algebra and Discrete Mathematics

This paper introduces the notion of almost positive posets as non-negative ones that contain maximal positive subposets. Such posets include both positive posets and \(P\)-critical posets (minimal non-positive ones) which were described by the authors back in 2005. Almost positive posets also include principal posets in the sense of D. Simson. By definition, a non-negative poset \(S=\{1,\cdots, n;\, \preceq\}\) is principal if the kernel of its Tits quadratic form \(q_S(z)=q_S(z_0,z_1,\cdots,z_n)\), defined by the equality \({\rm Ker}\,q_S(z):=\{t\in \mathbb{Z}^{1+n}\,|\, q_S(t)=0\}\), is an infinite cyclic subgroup of \(\mathbb{Z}^{1+n}\). In 2019, the authors described all serial principal posets. This paper concludes the description of all almost positive posets.