Least Upper Bound property - A set $X$ is said to have this property if every non empty subset $A$ of $X$ which is bounded above has the least upper bound. (This does not imply the least upper bound must belong to $A$, which is my inference from the definition of least upper bound)
The definition for the Greatest Lower Bound property is analogous.
Let $X=(0,1]$ and hence $( X, \leq)$ is a partially ordered set where $\leq$ has the usual meaning.
Now clearly $X$ has the least upper bound property as its closed above.
But if we take $A=X\subseteq X$ then what is the greatest lower bound of $A$? How come then $X$ has the greatest lower bound property?