Classification of minimal and maximal non-serial positive posets

The finite posets with positive Tits quadratic form, which are called positive, are analogs of Dynkin diagrams. They were first described in 2005 by the authors. In particular, according to this result such a poset can be serial if it belongs to an infinite strictly increasing sequence of positive posets, or non-serial otherwise. In the following years the authors studied various classes of posets that are related to the Tits quadratic form. In this paper, positive posets are studied in more detail, namely in relation to their ordering. The main theorems classify all non-serial positive posets that are minimal or maximal. The case of serial posets are trivial: there are no maximal posets and all minimal posets are one-element. The number of non-serial minimal posets up to isomorphism and duality is 10, and the number of maximal ones is 66 (out of a total 108). Cite as: V. M. Bondarenko, M. V. Styopochkina, “Classification of minimal and maximal non-serial positive posets,” Prykl. Probl. Mekh. Mat., Issue 22, 5–12 (2024), https://doi.org/10.15407/apmm2024.22.5-12