- ...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.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

- ...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].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Wed Feb 28 13:55:46 HST 1996