# Poset vs Set and Some Notations

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:

1. 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 $$x \in (X, R)$$?

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 $$x$$.

• Wouldn't a set be an unordered pair? Therefore, an ordered pair wouldn't be a set?
– Erin
Feb 13, 2021 at 10:31
• Please have a look at the Wikipedia link to see in detail what an ordered pair is. Feb 13, 2021 at 10:32
• @PrisonMike: In set theory everything is a set. Feb 13, 2021 at 11:51

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 $$R$$.