What does it mean by "A total order,$T$, $⪯$ is said to be compatible with the partial order $R$ if $aRb$ implies a $⪯$ b"?

Question

In this definition:

A total order, $T$, $⪯$ is said to be compatible with the partial order $R$ if $aRb$ implies a $⪯$ b

---

There are 2 parts I am confused about.

The 1^st part: "with the partial order R if $aRb$ implies a $⪯$ b"

The 2^nd part: "A total order $⪯$ is said to be compatible"

---

Is the 1^st part referring to the ordered pairs in $T$ or $R$?

IE are they saying

For every $(a,b)$$∈$$T$, $(a,b)$$∈$$R$ such that a $⪯$ b? OR

every $(a,b)$$∈$$R$ is such that a $⪯$ b

---

As for the 2^nd part, I have no idea.

Which total order are they suddenly referring to?

Are they just saying there exists some total order?

---

Apologies if the question isn't clear, I tried to pinpoint what I am not clear about?

Answer 1

I never heard called that compatible.
From that definition,, it just means that, as a set of ordered pairs, $R$ is a subset of $\preceq$.
In this situation, it is common to say that $\preceq$ is an extension of $R$.

Related with that, and of possible interest, is the Szpilrajn extension theorem.

---

Regarding your conjectures, the second one, about the first part is the correct one, that is, if $a R b$ then $a \preceq b$ (but not necessarily the converse, unless $R$ is already a total ordering).