In [C Mclarty] we read
[Noether] project was to get abstract algebra away from thinking about operations on elements, such as addition or multiplication of elements in groups or rings. Her algebra would describe structures in terms of the relations between selected subsets (such as normal subgroups of groups) and homomorphisms. Noether aimed to organize algebra around homomorphism and isomorphism theorems for each type of structure. And she meant to organize all mathematics around this algebra.
I was unable to find references of definitions of algebraic structures in this spirit.
This program was achieved? There are references of definitions following Noether’s project?
Why N Bourbaki did not adopt it (for the definition of its pure structures of algebra)?