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 $\prod_{i<n} K_i$ into $\prod_{j<m} L_j$.