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(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

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  • $\begingroup$ (1) Yes and no. The statement "$S$ is a partially ordered set" is not ambiguous at all... it merely says that there is some relation on the set which we can give a symbol to and state that it is a partial order. It is ambiguous as to what elements are related to what other elements and ambiguous as to what specific pattern the partial order takes beyond what is required by the definition. For example... the natural numbers are a partially ordered set ordered by $\leq$, the usual less than or equal to sign. It is also partially ordered by $\mid$, the divisibility sign, clearly different. $\endgroup$ – JMoravitz Apr 26 '18 at 5:59
  • $\begingroup$ (2) See my first comment above. As for what the LUB property implies, it implies what is stated by the definition... that any scenario in which you have upper bounds of a collection of elements, there must be some element which is "least" among all upper bounds. For the usual $\leq$ and a finite number of elements, this would be the $\max$ operator.. that is the least upper bound of $\{4,6\}$ according to the $\leq$ order would be $6$. In the case of the partial order $\mid$, one would have the least upper bound would be the least common multiple, so for $4,6$ would be $12$. $\endgroup$ – JMoravitz Apr 26 '18 at 6:06
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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.

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