Sub-lattices and lattices.

Question

I have read in a textbook that $ \mathcal{P}(X) $, the power-set of $ X $ under the relation ‘contained in’ is a lattice. They also said that $ S := \{ \varnothing,\{ 1,2 \},\{ 2,3 \},\{ 1,2,3 \} \} $ is a lattice but not a sub-lattice. Why is it so?

Answer 1

The point of confusion is that a lattice can be described in two different ways. One way is to say that it is a poset such that finite meets and joins exist. Another way is to say that it is a set upon which two binary operations (called meet and join) are given that satisfy a short list of axioms. The two definitions are equivalent in the sense that using the first definition's finite meets and joins gives us the two binary operations, and the structure imposed by the second definition allows one to recover a poset structure, and these processes are inverse to each other.

So now, if $L$ is a lattice and $S\subseteq L$ then $S$ is automatically a poset, indeed a subposet of $L$. But, even if with that poset structure it is a lattice it does not mean that it is a sublattice of $L$. To be a sublattice it must be that for all $x,y\in S$, the join $x\vee y$ computed in $S$ is the same as that computed in $L$, and similarly for the meet $x\wedge y$. This much stronger condition does not have to hold. Indeed, as noted by Gerry in the comment, the meet $\{1,2\}\wedge \{2,3\}$ computed in $\mathcal P({1,2,3})$ is $\{2\}$, while computed in the given subset it is $\emptyset$. None the less, it can immediately be verified that the given subset is a lattice since under the inclusion poset, all finite meets and joins exist.