Michael Barr writes:
This reminds me of a speculation I have often had (although Saunders denied and he knew Birkhoff pretty well). In the 30s and 40s, the word "homomorphism" was regularly used but always meant surjective. By the late 40s and 50s people were talking about "homomorphism into" meaning
not
necessarily surjective. So groups had lattices of subgroups and lattices of quotient groups and Birkhoff invented lattice theory at least partly in the hope that the structure of those two lattices would tell you a lot about the structure of the group. I don't think this actually happened to any great extent. But I have wondered whether Birkhoff might instead have
Noether's `set theoretic foundations of group theory', where group axioms are based on a notion of coset decomposition rather than multiplication, seems to be much earlier (20s) attempt to the same: http://www.math.jussieu.fr/~leila/grothendieckcircle/mclarty2.pdf
Michael
Nikita.