I'm searching for an ordering principle which combines two (or more) preorders, akin to the lexicographic order, but not quite: I'm looking for an ordering principle that does not priorities, but instead allows for ties.
To illustrate: the lexicographic ordering of $A={a,b}$ and $A'={a',b'}$ would be $aa'<ab'<ba'<bb'$, whereas I looking for the order $aa' < ab' \sim ba' < bb'$ (edit: where $\sim$ is pre-order equivalence).
The latter makes sense if you think of the elements as e.g. answers, letting $a$ and $a'$ be the correct ones. Then the answer pairs $ab'$ $a'b$ are equally wrong/right.
I'm looking for a characterization of this principle or a name for it so I can look it up. Help will be much appreciated.
If you are looking for a partial order on a Cartesian product that requires the appropriate order (in the weak sense: less than or equal) to hold separately on each of the components of the product. this is called the product order. The ordering on each component may be total or partial; the product order will always be partial (except in trivial cases like a product with a singleton set).