Question

(a) Is every total ordering a lattice? Why or why not?

Answer 1

A partially ordered set $(L, ≤)$ is called a lattice if each two-element subset $\{a, b\} ⊆ L$ has supremum and infimum in $L$, denoted by $a ∨ b$ and $a ∧ b$, respectively.

If $(L, ≤)$ is a total ordering, then by definition, ${\displaystyle a\leq b}$ or ${\displaystyle b\leq a}$ for all $a,b\in L$. If ${\displaystyle a\leq b}$, then $a ∨ b=b\in L$ and $a ∧ b=a\in L$. If ${\displaystyle b\leq a}$, then $a ∨ b=a\in L$ and $a ∧ b=b\in L$.

Therefore, every total ordering is a lattice.