Question about the definition of the upper set

Question

As I understand it, a subset $L$ of a partially ordered set ($S, \preceq$) is called a down-set or lower set if for any $s \in L$ and $s' \preceq s$, we have $s' \in L$. Now, my question is can we have a $t \in$ $L$ such that $t$ is not related to $s$ by $\preceq$ (i.e., $t$ and $s$ are not comparable)?

Answer 1

Yes, this can certainly happen. That is the meaning of the word 'partial' in "partially ordered set".

For instance, for any set $A$, the power-set $\mathcal P(A)$ of all subsets of $A$ is a poset (i.e., partially ordered set) when ordered by inclusion. It $A$ has more than one element, then there exists (typically many) elements $B,C\in \mathcal P(A)$, such that neither $B\subseteq C$ nor $C\subseteq B$.

Answer 2

The answer is yes. Just read the definition that you've quoted correctly. Nowhere in the definition does it say that it must be the case for every pair of elements in the lower set that one precedes the other. If such a condition were necessary, it would be mentioned. Maths is a rigorous subject.