Lattice Homomorphism

Let  and  be lattices, and let . Then  is a lattice homomorphism if and only if for any ,  and . Thus a lattice homomorphism is a specific kind of structure homomorphism. In other words, the mapping  is a lattice homomorphism if it is both a join-homomorphism and a meet-homomorphism.

If  is a one-to-one lattice homomorphism, then it is a lattice embedding, and if a lattice embedding is onto, then it is a lattice isomorphism.

An example of an important lattice isomorphism in universal algebra is the isomorphism that is guaranteed by the correspondence theorem, which states that if  is an algebra and  is a congruence on , then the mapping  that is defined by the formula

is a lattice isomorphism.

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