  In this paper, it is proved that, for the truth value algebra of
interval-valued fuzzy sets, the distributive laws do not imply the monotonicity
condition for the set inclusion operation. Then, a lattice-ordered
$t_{r}$-norm, which is not the convolution of $t$-norms on $[0, 1]$, is
obtained. These results negatively answer two open problems posed by Walker and
Walker in [15].
