Lattice homomorphism

Remarks

A lattice homomorphism is monotone: if $x \le y$ then $x \vee y = y$, so $f(x) \vee f(y) = f(y)$, that is, $f(x) \le f(y)$. A bijective lattice homomorphism is a lattice isomorphism, and its inverse is again a lattice homomorphism. A bijection between lattices is a lattice isomorphism exactly when it and its inverse are monotone. Example: every monotone map between two chains is a lattice homomorphism, since join and meet in a chain are $\max$ and $\min$, and monotone maps preserve both. Non-example: let $N = \set{\emptyset, \set{1}, \set{2}, \set{1, 2, 3}}$, a lattice under $\subseteq$ in which $\set{1} \vee \set{2} = \set{1, 2, 3}$. The inclusion map of $N$ into the power set of $\set{1, 2, 3}$ is monotone and preserves meets. It fails clause 1, because the join of $\set{1}$ and $\set{2}$ in the power set is $\set{1, 2} \ne \set{1, 2, 3}$.