Minimax sums of posets and the quadratic Tits form

Vitalij M. Bondarenko ; Andrej M. Polishchuk

Algebra and discrete mathematics, no. 1 (2004), pp. 17-36.

Résumé

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.

Détail

Mots-clés : poset, minimax sum, the rank of a sum, the Tits form.