On P-weight and P-distance inequalities

AbstractJang and Park asked in [On a MacWilliams type identity and a perfectness for a binary linear (n,n-1,j)-poset code, Discrete Math. 265 (2003) 85–104] whether, for each poset P={1,...,n}, the P-weights and P-distances satisfy the inequalities wP(x)-wP(y)⩽dP(x,y)⩽wP(x)+wP(y)-wP(xy) for all vectors x,y∈Z2n. We prove that these inequalities hold for all vectors x,y∈Fn over any field F