Extension results for boolean maps and a class of systems of linear inequalities

In this paper we introduce the notion of {\it core} for two specific classes of boolean maps on finite involution posets (which are a generalization of the boolean lattices) and we prove some extension results for such families of boolean maps. Through the properties of the core, we provide a complete characterization of such maps. The main purpose of such abstract results is their application to the study of the compatibility of a particular class of systems of linear inequalities related to a conjecture of Manickam, Mikl"os and Singhi (\cite{ManSin88}, \cite{ManMik87}), still unsolved and that can be considered dual to the theorem of Erd"os-Ko-Rado \cite{erd-ko-rad}.