A piecewise affine concave function $f$ on $N_{\mathbb{R}}$ is an H-lattice (respectively, a V-lattice) concave function if it has an H-representation (respectively, a V-representation) with integral coefficients. We say that $f$ is a rational piecewise affine concave function if it has an H-representation (or equivalently, a V-representation) with rational coefficients.