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

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