Let $(L,\ge)$ a partially ordered set. Suppose that for avery $S \subset L$, there exists an element $LUB(S)= a$ such that $x \ge a \iff x \ge u \quad \forall u \in S$ and, with the obvious meaning of $\le$, there exists an element $GLB(S)= b$ such that $x \le b \iff x \le u \quad \forall u \in S$.
Now: if $S=\emptyset$ is the emptyset, what are $LUB(\emptyset)$ and $GLB(\emptyset)$? Im not able to figure if this elements are defined and what they are. Someone can help my poor intuition?
Added after the comment:
This question comes from the definition of continuous geometry of von Neumann (see here ).
He define : $LUB(\emptyset)=\mathbf 0$ and $GLB(\emptyset)=\mathbf 1$ then says that we can verify ''immediately'' that:
$ \forall a \in L \; : \; \mathbf 0 \le a \le \mathbf 1 $
and: $ a \lor \mathbf 0 =a\;,\;a\land \mathbf 0=\mathbf 0\;,\;a\lor \mathbf 1=\mathbf 1 \;,\; a\land \mathbf 1=a $ where the symbols $\lor$ and $\land$ have the usual meaning for a lattice.
For me this result is not at all immediate, I suppose because I've not a good intuition of what are the elements $\mathbf 0$ and $\mathbf 1$.