**MSC:**- 06A06 Partial order, general
- 06A23 Complete lattices, completions

transitive relation \,$\le$\,, \,then the following completeness

properties are equivalent\,:

\renewcommand{\labelenumi}{(\arabic{enumi})}

\begin{enumerate}

\item $\operatorname{lb}\,\left(\0\{\,x\0, \,y\,\}\0\right)\ne \emptyset$ \ for all

\ $x\0, \,y\in X\0$, \,and \ $\inf\,(\0A\0)\ne \emptyset$ \

\ for all \ $A\subset X$ \ with \ $A\ne \emptyset$ \ and \

$\operatorname{lb}\,(\0A\0)\ne \emptyset$\,;

\item $\inf\,\left(\0\{\,x\0, \,y\,\}\0\right)\ne \emptyset$ \ for all

\ $x\0, \,y\in X\0$, \,and \ $\inf\,(\0A\0)\ne \emptyset$ \

\ for all \ $A\subset X$ \ with \ $A\ne \emptyset$\,, \

$\operatorname{lb}\,(\0A\0)\ne \emptyset$ \ and \ $\operatorname{ub}\,(\0A\0)\ne \emptyset$\,.

\end{enumerate}

Thus, we obtain a substantial generalization of a basic theorem of

Garrett Birkhoff which says only that in a conditionally complete

lattice every nonempty subset which has a lower bound has a greatest lower

bound.

OTKA T--043080 and the János Bolyai Scholarship of the Hungarian

Academy of Sciences.