Butterflies and Snakes

An important aspect of Linear Algebra, needed in practical applications, is based on the wider “Structure theorem for finitely generated modules over a principal ideal domain”.

This in turn can be generalized to “Goursat Categories” (or varieties), which include Maltsev varieties. I don’t really understand what this means, but prefer to understand the most general version as presumably closer to the “essence”.

Apparently this includes both the “Butterly Lemma” and the “Snake Lemma” (often referenced in category theory). Again, I don’t really understand what this is about yet, but curiously first came across something rather like both during transition from secondary school to early undergraduate mathematics or perhaps during first year. I still have a hardcopy of this book:

@book{papy1966groups,
title={Groups},
author={Papy, G.},
url={https://books.google.com.au/books?id=10DEjgEACAAJ},
year={1966},
publisher={Macmillan}
}

also available at:

https://archive.org/details/Groups_492

Papy-Groups_text.pdf

Chapter 10 describes “Groups with operators” which generalize vector spaces, modules and ideals. Operators are simply endomorphisms but not necessarily with a ring morphism or even a group morphism to endomorphism ring of the group. It includes proof of a structure theorem for “admissible” normal subgroups (compatible with the operators), based on “ladders” (suggestive of a snake diagram) from Schreier, with a “four sets lemma” from Zassenhaus (suggestive of a butterfly diagram).

I also have hardcopy of Brian Davey on Lattice Theory (3edn) which explains that “Fundamental Theorem of Arithmetic” is really a result concerning existence and uniqueness of join irredundant decomposition in distributive ACC lattices.

My impression is that the essence of what is needed can be stated in terms of ACC modular lattices and is applicable to a much wider range of structures than modules over a principal ideal domain – including the arithmoi and division rings as “lineal” spaces.

Collection of references below are for studying this.

@book{denecke2013galois,
title={Galois Connections and Applications},
author={Denecke, K. and Ern{\’e}, M. and Wismath, S.L.},
isbn={9781402018985},
series={Mathematics and Its Applications},
url={https://books.google.com.au/books?id=shfnBwAAQBAJ},
year={2013},
publisher={Springer Netherlands}
}

(Google search for “Goursat variety” found reference in above on p160)

Below has a college teacher’s account of exercises leading to Goursat’s lemma recommended for junior undergraduate courses. This may be a good place to start.

@article{Josep-2009,
doi = {10.2307/25653685},
title = {Goursat’s Other Theorem},
author = {Joseph Petrillo},
publisher = {Mathematical Association of America},
journal = {College Mathematics Journal},
issnp = {0746-8342},
issne = {1931-1346},
year = {2009},
month = {03},
volume = {40},
issue = {2},
page = {119–124},
url = {http://gen.lib.rus.ec/scimag/index.php?s=10.2307/25653685},
}

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