Ordered sets and the least upper bound (1)  When we say $S$ is a partially ordered set (POSET) is this somewhat ambiguous unless we refer to the partial order we have in mind?  I.e. isnt is possible that S may be ordered differently according to our order relation?
(2) When we say $S$ has the least upper bound property do we also face the same issue as above?  e.g. for real numbers with least upper bound property, what is really meant is least upper bound property with respect to less than or greater than.
EDIT: DELETED Q3
 A: When we say that $S$ is a poset, we usually are already referring to a specific order relation, that is, we actually mean $\langle S, \leq \rangle$ is a poset, where $\leq$ is any binary relation defined on $S$ which is reflexive, anti-symmetric and transitive.
So the partial ordering is often understood from the context, or from a previous reference to it.
The user JMoravitz gives in a comment a good example of two simple partial orderings on $\mathbb N$.
The idea that we mean that there can be defined a partial ordering on that set, I think it doesn't make much sense, because it is vacuous: for any set $S$, we can always define the poset $\langle S,=\rangle$.
It's an anti-chain, but a poset nonetheless.
It is also always possible to define a chain in any set (just take the anti-chain above, or any other ordering, and apply Szpilrajn Theorem).
When we say that a subset of $S$ has a least upper bound (or greatest lower bound) we are, of course, doing it with reference to the understood partial ordering.
