Lattice Homomorphism

Let L=<L, v , ^ > and K=<K, v , ^ > be lattices, and let h:L->K. Then h is a lattice homomorphism if and only if for any a,b in L, h(a v b)=h(a) v h(b) and h(a ^ b)=h(a) ^ h(b). Thus a lattice homomorphism is a specific kind of structure homomorphism. In other words, the mapping h is a lattice homomorphism if it is both a join-homomorphism and a meet-homomorphism.

If h 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 A is an algebra and theta is a congruence on A, then the mapping h:[theta,del _A]->Con(A/theta) that is defined by the formula

 h(phi)=phi/theta={([a]_theta,[b]_theta) in (A/theta)^2|(a,b) in phi}

is a lattice isomorphism.

See also

Lattice, Lattice Embedding, Lattice Isomorphism, Structure Homomorphism

This entry contributed by Matt Insall (author's link)

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Lattice Homomorphism

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Insall, Matt. "Lattice Homomorphism." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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