Reference Request for "Finite Semilattice with Top and Bottom is a Lattice"

Let $$\mathcal{O}(P)$$ be a finite, completely distributive lattice of all lower sets ordered by set inclusion. Moreover, let $$K =\; \mathrel{\{} h(x) \mathrel{|} x \in \mathcal{O}(P) \mathrel{\}}$$ be another poset ordered by set inclusion with $$h:\mathcal{O}(P)\rightarrow K$$ being a surjective function that is order- and join-preserving (which makes $$K$$ a join-semilattice, right?).

As far as I understand, because of $$h$$ being order-preserving, there must be a top element $$h(\top)$$ and a bottom element $$h(\bot)$$. If understand this question correctly, $$K$$ must hence be lattice (where meets are defined in terms of joins as $$a\wedge b=\bigvee\mathrel{\{} c \mathrel{|} c \subseteq a \text{ and } c \subseteq b \mathrel{\}}$$).

What does this mean with respect to $$h$$? Can I somehow conclude that $$h$$ is meet-preserving as well (which would leave $$h$$ as a lattice homomorphism)?

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I found this mathoverflow question here.

As part of the answer there is the proposition that "[...] a finite meet-semi-lattice with a maximum element is a lattice". I suppose this holds dually for a join-semilattice with a minimal element.

Is there any citable reference which proves this well-known proposition?

Background: I got a finite, completely distributive lattice (i.e., the set of all lower sets w.r.t a finite poset $$P$$ ordered by set inclusion; a superalgebraic lattice if I understand correctly), let us denote it as $$\mathcal{O}(P)$$. On top of that, I got a map $$h : \mathcal{O}(P) \rightarrow K$$ to another finite (semi-)lattice $$K$$ whose elements are sets ordered by inclusion as well. I already know that $$K$$ has a top $$h(\top)$$, a bottom $$h(\bot)$$ and that $$h$$ is join-preserving (which leaves $$K$$ at least as a join-semilattice, right?).

Now if I understand the aforementioned mathoverflow question correctly, this join-semilattice $$K$$ must be a lattice since there is a bottom. In particular, I am trying to prove that $$h$$ is meet-preserving where meets are defined in terms of joins as $$a\wedge b=\bigvee\,\{ \, c \;|\; c \subseteq a \textit{ and } c \subseteq b \,\}$$.

• But your "background" paragraph confuses me. In particular, it is unclear there what you are trying to show, versus what you know. Having a poset map into $K$ cannot help you prove that $K$ is a lattice, at least not without more information about the map than you have provided (e.g., that it is surjective). Commented Oct 6, 2023 at 14:17
• Anyways, for a reference to the fact you want, see Proposition 3.3.1 of Stanley's textbook "Enumerative Combinatorics," Volume 1. Commented Oct 6, 2023 at 14:18
• Thank you for the quick response and the reference. Defining the meet as $a \wedge b = \vee\{c:c\leq a, b\}$ is indeed clear to me. I think my confusion comes from me not knowing enough about $h$ myself yet (as you pointed out as well). The poset $K$ is itself defined using $h$ (i.e., $K=\{ h(x) \mathrel{|} x \in \mathcal{O}(P) \}$) where $h$ is order-preserving and surjective but not injective. Commented Oct 6, 2023 at 14:36
• It's not automatically true that a join semilattice homomorphism of finite lattices preserves meets Commented Oct 6, 2023 at 14:37
• @SamHopkins, my impression from the background of the question and the comments is the op wants to determine if h preserves meets and without more info on h we can't say anything Commented Oct 6, 2023 at 14:40

Okay, let me try to summarize.

The fact that a finite join semilattice with a minimum is a lattice is Proposition 3.3.1 of Stanley, "Enumerative Combinatorics," Volume 1.

Now suppose you have a finite lattice $$L$$ (corresponding to your $$\mathcal{O}(P)$$, but the fact that it is distributive is irrelevant), and another poset $$K$$ for which you have an order-preserving, join-preserving, surjective map $$h\colon L \to K$$. This does imply that $$K$$ is a join semilattice (for $$a,b \in K$$, we define the join $$a\vee b$$ to be $$h(a' \vee b')$$ where $$a', b' \in L$$ are pre-images under $$h$$ of $$a,b$$). Furthermore, the image of the minimum of $$L$$ under $$h$$ must be the minimum of $$K$$. And clearly $$K$$ is finite. So indeed, we can conclude that $$K$$ is a lattice.

EDIT: In response to the further question about whether $$h$$ must be meet-preserving, very simple examples show this is not the case. For instance, take $$L$$ to be the rank $$2$$ Boolean lattice of subsets of $$\{1,2\}$$, and $$K$$ to be the two element lattice $$\hat{0} < \hat{1}$$, with $$h(\{1\})=h(\{2\})=h(\{1,2\})=\hat{1}$$ and $$h(\varnothing)=\hat{0}$$. This $$h$$ is order-preserving, join-preserving, and surjective, but not meet-preserving.

• Thank you very much for this explanation and your time. On top of $K$ being a lattice, can we conclude anything about $h$? Provided the information we have, is there a way to show that $h$ is meet-preserving? Commented Oct 6, 2023 at 15:27
• @Björn: See my edit. We cannot conclude that $h$ is meet-preserving. Commented Oct 6, 2023 at 15:33
• The (more general) fact that a complete meet semilattice with a maximum (or dually a complete join semilattice with a minimum) is a complete lattice is Theorem 2 in Chapter IV of Birkhoff, Lattice Theory, 2nd edition.
– bof
Commented Oct 7, 2023 at 3:41