On \(P\)-weight and \(P\)-distance inequalities (Q2488934)

Jang 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, No. 1--3, 85--104 (2003; Zbl 1071.94023)] whether, for each poset \(P=\{1,\dots,n\}\), the \(P\)-weights and \(P\)-distances satisfy the inequalities \(w_P({\mathbf x})-w_P({\mathbf y}) \leq d_P(\mathbf{x,y})\leq w_P(\mathbf x)+w_P(\mathbf y)-w_P(\mathbf{xy})\) for all vectors \(\mathbf{x,y}\in\mathbb Z^n_2\). We prove that these inequalities hold for all vectors \(\mathbf{x,y}\in\mathbb F^n\) over any field \(\mathbb F\).