  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}.
