Is every chain with strict order a lattice?

Question

If P is a totally ordered set (chain) with strict order $\prec$ .. can we say P is a lattice ?. I mean we won't be having reflexivity at our disposal .. can we get over with it ? Can we show it otherwise ..

Answer 1

Yes. An order $(L, \leq)$ is a lattice if for every two elements $a, b \in L$, $\exists \sup\{a,b\}$ and $\inf\{a,b\}$. In a total strict order, as for instance a subset of $\mathbb{N}$, we can guarantee that the two previous conditions hold.