Minimax sums of posets and the quadratic Tits form

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.