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What is the most complete book on algebraic structures that deals with the complete taxonomy from magmas to Lie algebras and inner product spaces?

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    $\begingroup$ I've never seen such a thing, at least not as complete as you seem to be requesting. Browsing books.google.com, I see a great many books that don't get beyond groups and rings. The main problem is that it's hard to fit both the taxonomy and statements of much substance about them in the same book. Besides, the idea of "the complete taxonomy" is ludicrous. It's not as if the theory is history, you know. There are LOTS now and probably more being invented as we speak. Of course, if you pick any collection you're interested in, you will find dozens of in-depth books. So why not do that? $\endgroup$
    – rschwieb
    Commented Aug 9, 2018 at 18:40
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    $\begingroup$ Deals with it how? There are online lists and charts of some structures, and within some similarity types, pictures of what is known about the lattice of equational theories (varieties), but it would be less useful than a phone book and more massive unless you had a particular goal for such a classification. Gerhard "What Structure For This Compendium?" Paseman, 2018.08.09. $\endgroup$
    – Gerhard Paseman
    Commented Aug 9, 2018 at 18:42
  • $\begingroup$ The theory of operads covers lots of these structures -- though not all. Loday and Vallette have a book on them. $\endgroup$
    – darij grinberg
    Commented Aug 9, 2018 at 20:16
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    $\begingroup$ Try my book: Combinatorial algebra: syntax and semantics. There are many algebraic structures (from magmas to Lie algebras) with many connections between them, and it contains many deep theorems. It is published by Springer. $\endgroup$
    – user6976
    Commented Aug 9, 2018 at 22:37

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I don't believe such a taxonomy exists or is even possible in any useful sense. But it sounds as if you're looking for the subject of universal algebra, which studies general algebraic structures in a systematic way. Many books on that subject are available, including the ones listed at the bottom of the Wikipedia page just linked.

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    $\begingroup$ In my dissertation work I dealt with all similarity types which were finite and lacking constants, which covers a lot, and was useful to me. However, I suspect the poster is looking for a catalogue of structures and structure classes, and relations between them, and not looking for the methods or questions to ask. Gerhard "Should There Be An Answer?" Paseman, 2018.08.09. $\endgroup$
    – Gerhard Paseman
    Commented Aug 9, 2018 at 18:48
  • $\begingroup$ Don't be shy, link to your dissertation! $\endgroup$
    – Tom Leinster
    Commented Aug 9, 2018 at 18:57
  • $\begingroup$ OK. Gerhard "Guess I'd Better Finish It" Paseman, 2018.08.09. $\endgroup$
    – Gerhard Paseman
    Commented Aug 9, 2018 at 19:16

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