Question

Suppose one has an inclusion $\iota : D \hookrightarrow S$ where $D$ is a finite distributive lattice and $S$ is a finite join-semilattice.

If $\iota$ preserves all meets and joins one can show that $|J(D)| \leq |J(S)|$ i.e. $D$ has no more join-irreducibles than $S$.

Does this also hold if $\iota$ only preserves finite joins?

I only know that it is true in the special cases where $S$ is a distributive lattice or $S \cong S^{op}$ is order-dual.

Answer 1

Well, I proved that it is true with a friend yesterday, so I'll include it here.

Recall that we have an inclusion $\iota : D \hookrightarrow S$ where $S$ is a finite join-semilattice and $D$ is a finite distributive lattice. More explicitly: $S$ has all finite joins, so it is a complete lattice with a top and bottom element and furthermore $D$ has a top and bottom too.

In the case where $\iota$ preserves all meets then the same function defines a meet-semilattice morphism $\iota : D^{op} \hookrightarrow S^{op}$, so by duality we have the surjection $\iota' : S \twoheadrightarrow D$, hence $|J(D)| \leq |J(S)|$ because join-irreducibles (we exclude zero) in join-semilattices form the minimal generating set.

But the question was whether the same is true when $\iota$ preserves all joins.

First recall that in any finite distributive lattice $D$, the cardinality $|J(D)|$ is the size of the maximal chain in $D$ (counting the edges). Since we have an embedding $\iota : D \hookrightarrow S$, it follows that $S$'s maximal chain has size more than or equal to $n = |J(D)|$. For a contradiction assume that $|J(S)| < n$. Then $S$ is generated by $n-1$ elements, so we have a join-semilattice morphism $f : 2^{n-1} \twoheadrightarrow S$. By duality there is an embedding $f' : S^{op} \hookrightarrow (2^{n-1})^{op}$, hence $S^{op}$'s maximal chain has size at most $n-1$, as does its order-dual $S$, this being a contradiction.