On the Posets % MathType!Translator!2!1!AMS LaTeX.tdl!TeX -- AMS-LaTeX!% MathType!MTEF!2!1!+-% feaaeaart1ev0aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX% garmWu51MyVXgatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wz% aebbnrfifHhDYfgasaacH8qrps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeWaeaaakeaada% qadaqaaiabjAr8xnaaDaaaleaacaaIYaaabaGaam4AaaaakiaacYca% cqGH8aapaiaawIcacaGLPaaaaaa!3E9D! $${\left( {{\user1{\mathcal{W}}}^{k}_{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 theAs a consequence of the general theory, the logical dependency among inequalities is adequately captured by the sodefined posets W k 2 ; < À Á . 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 P k W k 2 È É 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. (2002): 06A05, 06A07, 13P10. Mathematics Subject Classification