Given a product order $\langle X_1 \times X_2 \times ... \times X_p, \preceq\rangle$ of ordered sets $\langle X_1, \preceq^\ast \rangle$, $\langle X_2, \preceq^" \rangle$, ...
Is there a possibility to deduce whether these ordered sets ($\langle X_1, \preceq^\ast \rangle$, $\langle X_2, \preceq^" \rangle$, ...) are total orders from the properties of the product order?
Let $A$ and $B$ be ordered sets with $b$ in $B$, give $A\times B$ the product order and let $p:A\times B \to A$, $(x,y) \mapsto x$ be the first projection.
Show $p(A\times\{b\})$ is order isomorphic to $A$.
Thus knowing the product order, one knows the order of the components.