On the Posets (W2k, <) and their Connections with Some Homogeneous Inequalities of Degree 2

A class of ranked posets {(D h, )} 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 , which are related to systems of the form {xaxb ∗abcd xcxd : 0 ≤ a, b, c, d ≤ k, ∗abcd ∈ {<, >}, 0 < x0 < x1 < ... < xk}. As a consequence of the general theory, the logical dependency among inequalities is adequately captured by the so-defined posets (W 2 , <). These structures, whose elements are all the D 2 ’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 {Pk ⊆ W 2 } 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. Key-words: β-linearisation, compatible system, homogeneous inequality, logical consequence, monomial inequality, specular element. 2000 MSC: 06A07, 06A05, 13P10. ∗Dipartimento Me.Mo.Mat. , via A. Scarpa 16, 00161 Roma, Italia; vietri@ dmmm.uniroma1.it; http://www.dmmm.uniroma1.it/̃ vietri