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On lattice-ordered $t_{r}$-norms for type-2 fuzzy sets
Published 7 Apr 2020 in math.GM | (2004.03593v1)
Abstract: 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].
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