Lattice (order) - Morphisms of Lattices

Morphisms of Lattices

The appropriate notion of a morphism between two lattices flows easily from the above algebraic definition. Given two lattices (L, ∨_L, ∧_L) and (M, ∨_M, ∧_M), a homomorphism of lattices or lattice homomorphism is a function f : L → M such that

f(a∨_Lb) = f(a) ∨_M f(b), and

f(a∧_Lb) = f(a) ∧_M f(b).

Thus f is a homomorphism of the two underlying semilattices. When lattices with more structure are considered, the morphisms should 'respect' the extra structure, too. Thus, a morphism f between two bounded lattices L and M should also have the following property:

f(0_L) = 0_M, and

f(1_L) = 1_M .

In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.

Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see preservation of limits. The converse is not true: monotonicity by no means implies the required preservation of meets and joins, although an order-preserving bijection is a homomorphism if its inverse is also order-preserving.

Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism. Similarly, a lattice endomorphism is a lattice homomorphism from a lattice to itself, and a lattice automorphism is a bijective lattice endomorphism. Lattices and their homomorphisms form a category.