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 is 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.

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.

If one takes the reflexive order relation as fundamental, which is a good practice 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, then this is just the product order.

That is, if one understands an "order" to be the reflexive relation, which comes along with its defined strict relation, then one has the (reflexive) product order relation, and your relation is the strict order arising from that product order.

So one could call this the (strict) product order, understanding "order" to be the reflexive relation, which comes along with its derived strict order.

But as you note, it is not the product of the strict orders, and I view this as one more reason that we don't want the strict orders to be primary.

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 is 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.