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

Bandelt, H. H. "Tolerance Relations on Lattices." Bull. Austral. Math. Soc.23, 367-381, 1981.Birkhoff,
G. Lattice
Theory, 3rd ed. Providence, RI: Amer. Math. Soc., 1967.Burris,
S. and Sankappanavar, H. P. A
Course in Universal Algebra. New York: Springer-Verlag, 1981. http://www.thoralf.uwaterloo.ca/htdocs/ualg.html.Chajda,
I. and Zelinka, B. "Tolerances and Convexity." Czech. Math. J.29,
584-587, 1979.Chajda, I. and Zelinka, B. "A Characterization of
Tolerance-Distributive Tree Semilattices." Czech. Math. J.37,
175-180, 1987.Gehrke, M.; Kaiser, K.; and Insall, M. "Some Nonstandard
Methods Applied to Distributive Lattices." Zeitschrifte für Mathematische
Logik und Grundlagen der Mathematik36, 123-131, 1990.Grätzer,
G. Lattice
Theory: First Concepts and Distributive Lattices. San Francisco, CA: W. H.
Freeman, 1971.Grätzer, G. Universal
Algebra, 2nd ed. New York: Springer-Verlag, 1979.Grätzer,
G. General
Lattice Theory, 2nd ed. Boston, MA: Birkhäuser, 1998.Hobby,
D. and McKenzie, R. The
Structure of Finite Algebras. Providence, RI: Amer. Math. Soc., 1988.Insall,
E. "Nonstandard Methods and Finiteness Conditions in Algebra." Ph.D. dissertation.
Houston, TX: University of Houston, 1989.Insall, M. "Some Finiteness
Conditions in Lattices Using Nonstandard Proof Methods." J. Austral. Math.
Soc.53, 266-280, 1992.Insall, M. "Geometric Conditions
for Local Finiteness of a Lattice of Convex Sets." Math. Moravica1,
35-40, 1997.Schweigert, D. "Central Relations on Lattices."
J. Austral. Math. Soc.37, 213-219, 1988.Schweigert, D.
and Szymanska, M. "On Central Relations of Complete Lattices." Czech.
Math. J.37, 70-74, 1987.