Zoltán Boros, Árpád Száz
Finite and conditional completeness properties of generalized ordered sets
The paper is published: Rostocker Mathematisches Kolloquium, Rostock. Math. Kolloq. 59, 75-86(2005)
MSC:
06A06 Partial order, general
06A23 Complete lattices, completions
Abstract: In particular, we show that if \,$X$ \,is a set equipped with a
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.
Keywords: Generalized ordered sets, lower bound and infimum completenesses.
Notes: The research of Zoltán Boros has been supported by the grant
OTKA T--043080 and the János Bolyai Scholarship of the Hungarian