Generalization of results from specific algebraic theories to Universal Algebra 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:


*

*Is there a specific reason not to generalize those results?

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

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