Find a set which has GLB and LUB but there exists at least one subset which has no GLB and LUB

Question

GLB=greatest lower bound LUB=least upper bound Give one example of a set such that the GLB and LUB exist but there exists at least one subset which has no GLB and LUB.

Answer 1

Consider the set $\{42\}$. It has a greatest lower bound and a least upper bound (both of which are $42$), but its subset $\varnothing$ has neither.

Answer 2

Consider the partially ordered set $$P=\{1,2,3,5,12,18,72,108,1080\}$$ ordered by divisibility. The set $X=\{5,12,18\}$ has greatest lower bound $1$ and least upper bound $1080,$ but its subset $Y=\{12,18\}$ has neither a greatest lower bound nor a least upper bound in $P.$