Minimax sums of posets and the quadratic Tits form

Abstract

Let \(S\) be an infinite poset (partially ordered set) and \(\mathbb{Z}_0^{S\cup{0}}\) the subset of the cartesian product \(\mathbb{Z}^{S\cup{0}}\) consisting of all vectors \(z=(z_i)\) with finite number of nonzero coordinates. We call the quadratic Tits form of \(S\)  (by analogy with the case of a finite poset) the form \(q_S:\mathbb{Z}_0^{S\cup{0}} \to \mathbb{Z}\) defined by the equality \(q_S(z)=z_0^2+\sum_{i\in S} z_i^2 +\sum_{i<j, i,j\in S}z_iz_j-z_0\sum_{i\in S}z_i\). In this paper we study the structure of infinite posets with positive Tits form. In particular,  there arise posets of specific form which we call minimax sums of posets.