SCHWEITZER 1972 Problem 2

Let $\leq$ be a reflexive, antisymmetric relation on a finite set $A$ . Show that this relation can be extended to an appropriate finite superset $B$ of $A$ such that $\leq$ on $B$ remains reflexive, antisymmetric, and any two elements of $B$ have a least upper bound as well as a greatest lower bound. (The relation $\leq$ is extended to $B$ if for $x,y \in A , x \leq y$ holds in $A$ if and only if it holds in $B$ .)

E. Freid