On the Posets ${\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)}$ and their Connections with Some Homogeneous Inequalities of Degree 2

Abstract

A class of ranked posets {(D _h ^k, ≪)} has been recently defined in order to analyse, from a combinatorial viewpoint, particular systems of real homogeneous inequalities between monomials. In the present paper we focus on the posets D ₂ ^k, which are related to systems of the form {x _a x _b * _abcd x _c x _d : 0 ≤ a, b, c, d ≤ k, * _abcd ∈ {<, >}, 0 < x ₀ < x ₁ < ...< x _k}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets ${\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)}$. These structures, whose elements are all the D ₂ ^k's incomparable pairs, are thoroughly surveyed in the following pages. In particular, their order ideals – crucially significant in connection with logical consequence – are characterised in a rather simple way. In the second part of the paper, a class of antichains ${\left\{ {{\user1{{\wp }}}_{k} \subseteq {\user1{\mathcal{W}}}^{k}_{2} } \right\}}$ is shown to enjoy some arithmetical properties which make it an efficient tool for detecting incompatible systems, as well as for posing some compatibility questions in a purely combinatorial fashion.