The classification of serial posets with the non-negative quadratic Tits form being principal

Abstract

Using (introduced by the first author) the method of (min, max)-equivalence, we classify all  serial  principal posets, i.e. the posets \(S\) satisfying the following conditions: (1) the quadratic Tits form \(q_S(z)\colon\mathbb{Z}^{|S|+1}\to\mathbb{Z}\) of  \(S\) is non-negative; (2) \({\rm Ker}\,q_S(z):=\{t\,|\, q_S(t)=0\}\) is an infinite cyclic group (equivalently, the corank of the symmetric matrix of \(q_S(z)\) is equal to \(1\)); (3) for any \(m\in\mathbb{N}\), there is a poset \(S(m)\supset S\)  such that \(S(m)\) satisfies  (1), (2) and \(|S(m)\setminus S|=m\).