Poset vs Set and Some Notations

Question

I apologize if I am splitting hairs here.

1. First of all, is a poset a set?

The official definition on my textbook says,

If $X$ is a set and $R$ is an ordering on $X$, the pair $(X, R)$ is called an ordered set.

I mean a poset is a set with a partial order, whereas a set itself, by definition, does not have order. So I guess a poset is not a set? But if this is true, then I have another confusion:

2. There's a line from the book that says,

A finite linearly ordered set has a (unique) largest element.

So if $x \in X$ and this $x$ is this largest element, is it ok to also write $x \in (X, R)$?

Answer 1

A poset is the combination (ordered pair) of a set and a partial order $R$ on that same set.

So you could say that a poset $(X,R)$ has underlying set $X$ and an order $R$. Often the order is understood from context and we just say the poset $X$, which is not entirely correct formally. A poset is a set with extra structure, like a group, or a field, or a topological space, etc.

If a poset has a maximum, just say $x \in X$, not $x \in (X,R)$. It is an element of the underlying set, with special properties wrt the order $R$.

Answer 2

1. $(X, R)$ is an ordered pair. Which in set theory is a set.

2. You have $x \in X$. But you can’t say $x \in (X,R)$ as $(X,R)$ is indeed a set, but not a set that contains $x$.