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I'm relatively new to universal algebra, but it seems that lots of theorems from specific algebraic theories (groups, rings) can be stated in the context of universal algebra, perhaps I'm wrong.

Examples being Jordan-Holder, Lasker-Noether, Hilbert Nullstellensatz (maybe not?) and many more, where one can talk about sobobjects, and congruences instead of "subgroups" and "ideals".

Sometimes (obviously) those theorems fail in the universal context, but some of the results stand, and even more so: We might be able to measure the failure of those theorems in the universal context. I hardly see those kind of deep results in introductions to universal algebra even though they can give a strong "justification" for the study of the field. Therefore I have a few questions:

  1. Is there a specific reason not to generalize those results?

  2. Is there a reason such results are not thought in Universal Algebra introductions (at least all the ones I found)?

  3. Is these sort of generalizations have open research problems that cause difficulties (relates to the first question)?

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  • $\begingroup$ By the way, which introductions have you read? Gerhard "May Have You Meet Others" Paseman, 2018.02.06. $\endgroup$
    – Gerhard Paseman
    Commented Feb 6, 2018 at 19:25
  • $\begingroup$ Google searches for introductory text all reveal text with basically the same content organized a bit differently, for example: Sankappanavar, H.P. and Burris, S., 1981. A course in universal algebra. Graduate Texts Math, 78. $\endgroup$
    – Omer Rosler
    Commented Feb 6, 2018 at 19:33
  • $\begingroup$ So it sounds like the answer is one. I recommend Graetzer, Whitehead, Birkhoff, McKenzie McNulty and Taylor, the short course by Jipsen, and some writings of Kearnes and of Willard. And this is just for introductory and overview material, not yet for indepth study. Check those out. Gerhard "And Come Back For More" Paseman, 2018.02.06. $\endgroup$
    – Gerhard Paseman
    Commented Feb 6, 2018 at 19:46
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    $\begingroup$ The following thread goes in the opposite direction, but I think it might be interesting to you. $\endgroup$
    – Pedro Sánchez Terraf
    Commented Feb 11, 2018 at 16:23
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    $\begingroup$ @Pedro Sanchez Terraf Actually, these counterexamples are of interest in the sense that if we look at them as pathologies, the properties we require from a universal algebra to prevent them would lead to a more fundamental understanding of the "regularity" of the classical algebraic theories. In analogy, the generalization from the integers to number rings caused the failure of unique factorization, which leaded to the theory of ideals and Dedekindness for a unified framework. $\endgroup$
    – Omer Rosler
    Commented Feb 11, 2018 at 18:38

3 Answers 3

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One of the reasons for the richness of group theory is that the structure represents automorphisms of another structure, and through conjugation gives rise to (sometimes trivial) automorphisms of its own structure. Not every structure can represent automorphisms of some object. A similar reason for the richness of universal algebra is that lattices are structures that represent information about a structure. You can't perform the same representation by using an arbitrary structure that is not a lattice.

Because of the intrinsic nature of the theorems involved (and because of the wide range of structure behaviour), you won't always find a nice generalization of a group theoretic result to arbitrary structures. One example is the cyclic nature of a group: you can't always find a way to "iterate" to an identity element from an arbitrary element (by self powers or inverse) because not every structure has an identity element. So one has to pick the properties and the range of applicable structures with care.

While generalizing known results are one motivation (and I think the homomorphism theorems are ideal in this respect), the subject of universal algebra has its own character apart from group theory or ring theory or even lattice theory, and it should be introduced as what it is: a means to study many different types of structures, not just as a venue for your questions 1 and 3.

Gerhard "That's How I Was Taught" Paseman, 2018.02.06.

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    $\begingroup$ This seems mostly true, but some of the results for groups/rings are about the algebraic structure and even the geometric reasoning (i.e. symmetry) can be phrased in Universal algebra as well. Decomposition series, ideal theory, radicals and even polynomials (although this generalization is much more involved) is in a sense the same theory of pure algebra $\endgroup$
    – Omer Rosler
    Commented Feb 6, 2018 at 15:25
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    $\begingroup$ Then write a position paper on this. While I do not share your perspective, I think it is a valuable one to explore. The blocks you will encounter in expressing this perspective will be a better answer to your question than any I can give. Gerhard "Let Nature Be Your Guide" Paseman, 2018.02.06. $\endgroup$
    – Gerhard Paseman
    Commented Feb 6, 2018 at 16:14
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A Google search immediately brings a few results including Master thesis of Junpei Sekino from 1969 where a generalisation of the Jordan-Hölder theorem for sufficiently arbitrary algebraic structures is proved, and some other classical theorems are also generalised. So I suppose your questions 1 and 3 are based on a slightly incorrect assumptions. As for question 2, even though I work a lot with rather general algebraic structures myself, my motivation is usually extrinsic so I am not in the position to talk for people who specialise in universal algebra due to intrinsic motivations (who probably wrote the introductory textbooks you are talking about).

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    $\begingroup$ Thank you for the reference, it seems I missed this one. I now found more of these results scarced around the internet, it seems indeed I am misinformed. Regarding the motivation, I guess it really depends and I hope an expert with intrinsic motivation can shed some light on this question $\endgroup$
    – Omer Rosler
    Commented Feb 6, 2018 at 18:59
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Many of the results you mention form part of the categorical approach to (parts of) Universal Algebra. There is a lot of active research in this area, especially in the study of semi-abelian categories, Malt'sev categories, etc. Although this is not main stream Universal algebra and some universal algebraists may not feel it deserves mention as part of that subject, it may help to look at the book by Bourn and Borceux mentioned in the n-Lab article on semi-abelian categories.

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