Geodesic
Minimax sums of posets and the quadratic Tits form
Algebra and discrete mathematics, no. 1 (2004), pp. 17-36

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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 $q_S(z)=z_0^2+\sum_{i\in S} z_i^2 +\sum_{i$. 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.
Keywords: poset, minimax sum, the rank of a sum, the Tits form. 
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     title = {Minimax sums of posets and the quadratic {Tits} form},
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Vitalij M. Bondarenko; Andrej M. Polishchuk. Minimax sums of posets and the quadratic Tits form. Algebra and discrete mathematics, no. 1 (2004), pp. 17-36. http://geodesic.mathdoc.fr/item/ADM_2004_1_a2/