The finite product of connected linearly ordered sets cannot be embedded into the product of a smaller dimension - Ishiu, Tetsuya (Topology Proceedings, 2025)
We shall show that for every positive integers n and m with m<n, if K_i is a connected linearly ordered topological space with at least two points for all i<n and L_j is a connected linearly ordered topological space for all j<m, there exists no continuous injective function from a nonempty open subset of _i<n K_i into _j<m L_j.