What do you call this relation between pre-orders?

Question

Let $\sqsubseteq_1,\sqsubseteq_2$ be two pre-orders.
Say that $\sqsubseteq_2$ perfects $\sqsubseteq_1$ if:

• $a \sqsubset_1 b$ implies $a \sqsubset_2 b$, and
• if $a$ and $b$ are incomparable according to $\sqsubseteq_2$, they are incomparable according to $\sqsubseteq_1$

You can see that "perfecting" is a pre-order on pre-orders, whose top elements are linear orders. Notice that "perfecting" is incomparable with inclusion (aka relation refinement).

Is "perfecting" a standard notion?

I checked the two books called "Ordered Sets" (Harzheim and Schroeder) but I had no luck.

Answer 1

These are fully faithful functors if you view preorders as categories. (A preorder is the same as a category such that for any objects X and Y there is at most one morphism from X to Y.)