Examples of lattice homomorphism $f:L\to K$, where $L$ is arbitrary

Question

Definition. Let $L=\langle L, \vee , \wedge \rangle$ and $K=\langle K, \vee , \wedge \rangle$ be lattices, and let $h:L\to K$. Then $h$ is a lattice homomorphism if and only if for any $a$,$b \in L$, $h(a \vee b)=h(a) \vee h(b)$ and $h(a \wedge b)=h(a) \wedge h(b)$. (If the lattice is bounded, the homomorphism should preserve bounds).

I want to see some examples of lattice homomorphism $f:L\to K$, where $L $ is arbitrary. I know $L\to L / \theta$, where $\theta$ is a congruence relation on $L$.

Thank you

Answer 1

1. You already mentioned a class of examples, the natural homomorphisms from a lattice onto a quotient of that lattice by one of its congruence relations, that is, $\nu_{\theta}:L \to L/\theta$, where $$\nu_{\theta}(x) = [x]_{\theta}$$ is the equivalence (congruence) class of the element $x$. In this case, it's always a surjective homomorphism

2. Another class of examples is given by sublattices: If $K$ is a sublattice of $L$, then $\iota:K\to L$, where $$\iota(x)=x.$$ In this case, it is always an injective homomorphism.

3. I'll give you another class of examples.
To keep it simple, I'll do it just for finite lattices; in this case, finite distributive lattices.
Let $P$ and $Q$ be finite posets and $\psi:P\to Q$ be order-preserving.

Notation. Let $\mathcal O(X)$, for a poset $X$, denote the lattice of the order-ideals (aka, down-sets) of $X$. Indeed, this is a distributive lattice.

So $\mathcal O(P)$ and $\mathcal O(Q)$ are finite distributive lattices.
Define $h_{\psi}:\mathcal O(Q) \to \mathcal O(P)$ (notice the reversing in the order of the arrows) by $$h_{\psi}(U) = \psi^{-1}(U) = \{p \in P : \psi(p) \in U\}.$$ Then $h_{\psi}$ is a lattice homomorphism (which also preserve the bounds: $h_{\psi}(\varnothing) = \varnothing$ and $h_{\psi}(Q) = P$).
(You can see more about these examples in Birkhoff's Representation Theorem. See also Introduction to Lattices and Order, by Davey and Priestley, chapter 5—Representation:the finite case)

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There are many examples of lattice homomorphisms which are not in any of these categories (for example, lattices homomorphisms which are neither injective nor surjective, and the lattices are not distributive), but it's not clear how many examples are enough to give you some insight.
Perhaps someone will give you some different classes of examples.

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Edit. (OP asked for a definition.)

Definition. Let $P$ be a poset and $U \subseteq P$. Then we say that $U$ is an order-ideal (or down-set) if for $u \in U$ and $p \in P$, if $p \leq u$ then $p \in U$.

So an order-ideal is a subset which is closed under taking "smaller" elements.
Sadly (from my point of view) the definition on Wikipedia asks that the subset must also be non-empty (see Ideal (order theory). It hads to the confusion the fact that in the same page, an order-ideal is defined as a "lower set", with a link to the page with Upper and Lower sets, where the empty set is allowed. That's the definition which is useful in Birkhoff representation.