I'm studying from Michael Carter's "Foundations", and on page 29 he makes the comment, "Note that the requirement of being a sublattice is more stringent than being a complete lattice in its own right".
This seems counterintuitive. Some lattice $L$ is complete if every nonempty subset $S\subseteq L$ has a least upper bound and greatest lower bound in $L$. This is a more stringent requirement than merely needing bounds for every combination of two points in $L$. By the same token, requiring that all subsets $T \subseteq S$ have meets and joins in $S$ (thereby making $S$ complete) ought to be more stringent than merely requiring $S$ to be a lattice, right? In what sense are the requirements for being a sublattice more stringent on $S$?
One more related question – suppose $X= \{ (1,1),(2,1),(3,1),(1,2),(1,3),(3,3) \}$. The book says $X$ is a complete lattice, but not a sublattice of $\{1,2,3\}\times \{1,2,3\}$. The point $(3,3)$ is an upper bound for all pairs of points in $X$, so shouldn't that be enough to qualify $X$ as a sublattice of the set above? Why is $X$ not considered a sublattice?
Best Answer
The meet and join operations in a sublattice must agree with the meet and join operations in the lattice. Consider your example. The least upper bound, in $X$, of $(1,2)$ and $(2,1)$ is $(3,3)$. In the space $\{ 1,2,3 \} \times \{ 1, 2, 3 \} $ the least upper bound of $(1,2)$ and $(2,1)$ is $(2,2)$.