Birkhoff's HSP theorem.
Given a class of algebraic structures which satisfies a set of identities (universally quantified equations in a purely functional language with equality as the only relation; this is known as an equational class), homomorphic images, images of subalgebras, and cartesian products also satisfy the same set of identities (e.g. by inspection) and their consequences in equational logic. Conversely, given a class of algebraic structures of the the same type closed under products, subalgebras, and homomorphisms, one can realize every such algebra as an image of some free algebra with respect to the class; constructing the appropriate image of the term algebra gives a free algebra that lives in the closed class; the (fully invariant) congruences on the term algebra gives the set of equations that the closed class must satisfy; thus a class is an equational class iff it is closed under the class operator HSP.
Garret
Garrett Birkhoff's original paper on this theorem (called a preservation theorem when it came out in the 1930's) is easy to read. There may be within it a cleaner synopsis than the one above.
Gerhard "Ask Me About System Design" Paseman, 2010.10.27
