# How can we express that a partial order is more complete than another one?

Suppose we have two partial orders $$R$$ and $$I$$ on $$\mathbb{C}$$ (conplex numbers) such that:

• $$R$$ is a total order on $$\mathbb{R}$$ (real numbers).
• $$I$$ is also a total order on $$\mathbb{R}$$ and, additionally, on $$\mathbb{I}$$ (imaginary numbers).

Is there a technical term fore expressing that $$I$$ is more complete than $$R$$ in the sense that it has more pairs in its domain?

Would there be a different terminology for the following cases?

• The case where the hierarchy defined by $$I$$ for $$\mathbb{R}$$ is the same as $$R$$'s.
• The case where the hierarchy defined by $$I$$ for $$\mathbb{R}$$ is different from $$R$$'s.

A partial order $$\leq^*$$ on a set $$X$$ is an extension of another partial order $$\leq$$ on $$X$$ provided that for all elements $$x$$ and $$y$$ of $$X,$$ whenever $$x \leq y,$$ it is also the case that $$x \leq^* y.$$ A linear extension is an extension that is also a linear (i.e., total) order.
Fortunately, in the case you're interested in (at least, in the first of the two cases you listed), $$I$$ is a linear extension of $$R.$$ So there is a well-established technical term that says exactly what you want.
It's not clear to me what would be meant by $$I$$ having (in your words) more pairs than $$R$$ when it is not a superset of $$R,$$ as seems to be implied by this sentence in the question:
The case where the hierarchy defined by $$I$$ for $$\mathbb{R}$$ is different [from] $$R$$'s.