What is the opposite of an "extension" of a partial order?

Question

Quoting Wikipedia:

A partial order ≤^* on a set X is an extension of another partial order ≤ on X provided that for all elements x and y of X, whenever x ≤ y, it is also the case that x ≤^* y.

So we can say:

≤^* is an extension of ≤

How can we phrase that in reverse? I'd like to say something like:

≤ is a _____ of ≤^*

Here's a quick example drawing to emphasize that ≤^* and ≤ may be over the same domain:

Answer 1

The word "restriction" refers to shrinking the domain. What you're after is refinement: $\le$ is a refinement of $\le^*$ if $a\le b\implies a\le^* b$, but not necessarily conversely.