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)?
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