Remark

A lattice is obviously both left and right directed. But an ordered set which is both left and right directed is not necessarily a lattice. \* An example of the latter is the set of mappings $x\mapsto p(x)$ of $\mathbf{R}$ into itself, where $p$ is a polynomial in $\mathbf{R}[X]$, this set being ordered by the relation $p\leqslant q$ (no. 4). \*