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.
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
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