I’ve had trouble finding a well-established term for the following very obvious and elementary construction on strict partial orders (i.e. transitive, irreflexive relations):
Given two strict partial orders $(X,<_X)$, $(Y,<_Y)$, there is a strict partial order on $X \times Y$ where $(x,y) < (x',y')$ just if one of the following cases holds:
- $x<x'$ and $y<y'$;
- $x=x'$ and $y<y'$;
- $x<x'$ and $y=y'$.
This corresponds clearly to the direct product of the (non-strict) partial orders $\leq_X$, $\leq_Y$ corresponding to $<_X$, $<_Y$. However, it’s not their direct/cartesian product as strict partial orders — or at least, it would be misleading to call it either of those, since those have another more obvious meaning. But presumably many other people must have had cause to make use of this product at some point or another. Does it have a well-established name?