Classification of the almost positive posets

Abstract

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.