= Homomorphism of lattices
A homomorphism between <lattices> preserves binary joins and meets: $f(x\vee y)=f(x)\vee f(y)$ and $f(x\wedge y)=f(x)\wedge f(y)$. It is order preserving, since $x\leq y$ implies $x=x\wedge y$ and hence $f(x)=f(x)\wedge f(y)$. For a map into the two-element lattice, the inverse image of zero is a <lattice ideal>, and meet preservation says this ideal is <prime>: if a meet belongs to it, at least one factor does. Conversely the indicator of the complement of a <prime> <lattice ideal> preserves both operations. Preservation of distinguished top and bottom elements is an additional requirement for homomorphisms of bounded lattices.
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