A partial order $(X, <_X)$ has order dimension $n$ if it can be realized as the product order of $n$ total orders, which means that there is an order-embedding between $(X, \lt_X)$ and $(Y^n, \lt)$ for some totally ordered set $Y$, where we define $\lt$ as $\bigwedge_i \lt_{i,i}$. (The $\lt_{i,j}$ are the orders on tuples defined by $x \lt_{i,j} y$ iff $x_i \lt y_j$).
A partial order $(X, <_X)$ is an interval order if it can be realized as the interval precedence order on some set of intervals, which means that there is an order-embedding between $(X, \lt_X)$ and $(\mathbb{N}^2, \lt)$ where we define $\lt$ as $\lt_{1,1} \wedge \lt_{1,2} \wedge \lt_{2,1} \wedge \lt_{2,2}$.
Is there some generalization of these notions which looks at other ways to define a partial order on tuples from the $<_{i,j}$?
