How is GLB and LUB different than the maximum and minimum of a poset? As the subject asks, how is the greatest lower bound different than the minimum element in a poset, and subsequently, how is the least upper bound different than the minimum? How does a set having no maximum but multiple maximal elements affect the existence of a LUB?

Edit:
Here's an assignment question I have on the topic...

Consider the subset $A=\{4,5,7\}$ of $W$.
Does $\sup(A)$ exist? What about $\inf(A)$?
I'm guessing that Sup is Suprema (or LUB), and inf is Infimum (GLB)... but I have no idea because I missed that day in class.
All I can guess is that the subset $A$ is ordered such that $7\preceq 5$, but $4$ and $5$ are incomparable. I don't know what to do with it.

The supremum coincides with maximum in the finite case. That said, a great deal of basic example are not finite, so supremum is a different operation in general. The most notorious thing here is that supremum is an element not necesarilly in the original set.
