On the Posets ( \user1Wk2 , < ){\left( {{\user1{\mathcal{W}}}^{k}_{2} , < } \right)} and their Connections with Some Homogeneous Inequalities of Degree 2

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 . 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 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.     

The Whitney embedding theorem for tropical torsion modules: Classification of tropical modules

We prove here a tropical version of the well-known Whitney embedding theorem [32] stating that a smooth connected m-dimensional compact differential manifold can be embedded into R^2m+1.The tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module , where p (equals to 2 for m=2, and otherwise) is the number of rows supporting the torsion, when the generators are given by the (independent) columns of a matrix of size n×m.As a corollary, we get that tropical m-dimensional torsion modules are classified by a (m-1)(m(m-1)-1)-parameter family.