...lattice
If f is a function with domain A, then f determines an equivalence relation on A. When f is a homomorphism this equivalence relation is called a congruence relation. The set of all congruence relations on A forms a lattice. If A is a group then this lattice is, of course, isomorphic to the lattice of normal subgroups.
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...monograph [13].)
A related story: Bjarni Jónsson used lattice theory to define quasi-isomorphism of Abelian groups and to show that, under this notion, the Krull-Schmidt Theorem on the uniqueness of direct decompositions, held for torsion-free Abelian groups of finite rank. In the mid-seventies Jónsson bemoaned to me that Fuchs, in the latest version of his book on Abelian groups, had completely removed any trace of lattice theory from his proof of this theorem. A month later I was talking with Lee Lady who told me that Jónsson did the Abelian group community a great favor by proving his theorem with lattice theory: it forced them to reformulate and reprove it and thereby understand it much better. See Chapter 3 of [15].
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Ralph Freese
Wed Feb 28 13:55:46 HST 1996