On lattice-ordered $t_r$-norms for type-2 fuzzy sets

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].