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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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.
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.
is an algebra and 
is a congruence on 
, then the mapping ![h:[theta,del _A]->Con(A/theta)](/images/equations/LatticeHomomorphism/Inline13.svg)
that is defined by the formula
![h(phi)=phi/theta={([a]_theta,[b]_theta) in (A/theta)^2|(a,b) in phi}](/images/equations/LatticeHomomorphism/NumberedEquation1.svg)
