A question about lattices and their being partially-ordered sets, rather than total-ordered sets

Question

Am I correct in saying that, given a and b are elements of S, the join(a, b) and meet(a, b) won't be either a or b, if there is no order between a and b?

Answer 1

(So the question doesn't remain without answers).

Yes, you are correct. If $a$ and $b$ are comparable, say $a\leq b$, then $a\land b = a$ and $a\lor b= b$; conversely, since $a\land b\leq a$ and $a\land b\leq b$ for all $a$ and $b$, if $a\land b\in\{a,b\}$, then $a$ and $b$ are comparable (similar if $a\lor b\in{a,b}$).