# Supremum for pair of elements in Hasse diagram

I know infimum is greatest lower bound, and supremum is least upper bound. I hope did this correctly. sup({3,4})=7

sup({4,5})=6

sup({3,1})=3

inf({5,1})=1

inf({8,7})=6

But what is inf({1,2}) and sup({8,7})? I mean two elements which aren't comparable.

## 2 Answers

The supremum or infimum of a subset of a poset does not necessarily exist: there might not be a least upper bound, or a greatest lower bound. For instance, the set $\{1,2\}$ has no lower bounds at all, so it can't have a greatest lower bound, so the infimum does not exist. Similarly, $\sup(\{8,7\})$ does not exist.

For a somewhat more interesting example, $\sup(\{1,2\})$ also does not exist. There are upper bounds of $\{1,2\}$ (namely $3,5,6,7,$ and $8$), but there is no least upper bound, since there is no single upper bound that is less than or equal to every upper bound. For instance, there is no upper bound that is both $\leq 3$ and $\leq 5$.

(In fact, two of your answers are incorrect, because you have given an upper or lower bound that is not the least upper bound or greatest lower bound. See if you can find which ones.)

The upper bounds of $\{3,4\}$ are $7$ and $8$, and neither of these are a least upper bound.

Likewise the, lower bounds of $\{8,7\}$ are $1,2,3,4,5,6$, and neither of those are greatest.