A morphism of distributive lattices is just a lattice homomorphism as given in the article on lattices, i . e . a function that is compatible with the two lattice operations.
12.
Similarly, a " lattice endomorphism " is a lattice homomorphism from a lattice to itself, and a " lattice automorphism " is a bijective lattice endomorphism.
13.
In duality of categories between, on the one hand, the category of finite partial orders and order-preserving maps, and on the other hand the category of finite distributive lattices and bounded lattice homomorphisms.
14.
With respect to the algebra of subsets, by the above we see that the inverse image function is a lattice homomorphism while the image function is only a semilattice homomorphism ( it does not always preserve intersections ).
15.
In bounded lattice " L " is called a "'0, 1-simple lattice "'if nonconstant lattice homomorphisms of " L " preserve the identity of its top and bottom elements.
16.
For example, lattice homomorphisms are those functions that " preserve " non-empty finite suprema and infima, i . e . the image of a supremum / infimum of two elements is just the supremum / infimum of their images.
