Left-continuity of -norms on the -dimensional Euclidean cube

Left-continuity of t -norms on the unit interval [ 0 , 1 ] is equivalent to the property of sup-preserving, but this equivalence does not hold for t -norms on the n -dimensional Euclidean cube [ 0 , 1 ] n for n ≥ 2 . Based on the concept of direct poset we prove that a t -norm on [ 0 , 1 ] n is left-continuous if and only if it preserves direct sups.