Terminology: product on strict preorders corresponding to direct product of preorders? 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?
 A: If one takes the reflexive order relation as fundamental, then this is just the strict product order. 
It is good practice to take the reflexive order relation as the primary relation, since in the context of pre-orders, one can define the strict order $<$ from the reflexive order $\leq$, but not necessarily conversely, since there are strict orders $<$ that arise from more than one pre-order. 
Thus, one understands an "order" to be the reflexive relation, which comes along with its defined strict relation. With two such orders, then one has the (reflexive) product order relation, and your relation is the strict order arising from that product order. 
So I would just call it the strict product order, meaning the strict order notion arising from the (reflexive) product order.
As you note, this is not the product of the strict orders, and I view this simply as one more reason that we don't want the strict orders to be the primary order notion. 
A: I believe that Paul Taylor calls this the "interleaved product" in his book Practical Foundations of Mathematics.
