Joins in lattices and sublattices

Question

Let $A$ be a lattice, and $B$ be a sublattice of $A$.

Why is the join of $A$ included in the join of $B$? That is, why is $\bigcup_{t\in T}^{A} a_t\leq\bigcup_{t\in T}^{B} a_t$?

(I am tempted to think the opposite inclusion, i.e. the join of $B$, as a sublattice, is included in the join of $A$).

Answer 1

If a subset $B$ of a lattice $A$ is a sublattice, then by definition all joins in $B$ are the same whether they are taken in $B$ or $A$. A sublattice is the same thing as an injective homomorphism that preserves meet and join. It's true though that arbitrary meets and joins may not be preserved.

If we consider an arbitrary subset that is a lattice with possibly different meet and join, the join in $B$ will always be larger than the join in $A$ because the join in $A$ is by definition less than or equal to every element that is greater than or equal to every element in the join. This also applies for infinite lattices. The join is the smallest element dominating all of the given elements.

Answer 2

Here’s a concrete example. Let $A$ be $\langle\Bbb R,\le\rangle$, and let $B$ be $\langle[0,1)\cup\{2\},\le\rangle$. Then $\bigvee^A[0,1)=1$, but $\bigvee^B[0,1)=2$. (Of course in both lattices join and meet are simply $\max$ and $\min$.)