Real subset sums and posets with an involution

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let (X,≤,c) be a finite poset, where c : X → X is an order- reversing and involutive map such that c(x) ̸= x for each x ∈ X. Let B2 = {N < P} be the Boolean lattice with two elements and W+(X,B2) the family of all the order- preserving 2-valued maps A : X → B2 such that A(c(x)) = P if A(x) = N for all x ∈ X. In this paper, we build a family Bw+(X) of particular subsets of X, that we call w+-bases on X, and we determine a bijection between the family Bw+(X) and the family W+(X,B2). In such a bijection, a w+-basis Ω on X corresponds to a map A ∈ W+(X,B2) whose restriction of A to Ω is the smallest 2-valued partial map on X which has A as its unique extension in W+(X,B2). Next we show how each w+-basis on X becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.