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I have a question which I'd like to ask, but I'm not sure it's appropriate for MO, and I wanted to get some feedback.

The reasons I'm worried it might be inappropriate are: (1) there are already multiple "counterexamples in ---" questions on MO, and presumably there's a law of diminishing returns on these; (2) general hesitation about soft/big list questions.

Tentative text of question below:

(Tentative tags: big-list, universal-algebra, counterexamples, lattices. "Soft question" doesn't seem appropriate, and even though "examples" is a far more common tag, "counterexamples" seems much more appropriate. I threw in the "lattices" tag because I suspect that many counterexamples will be about lattices somehow, and I'd like to attract attention from lattice theorists. I'm not sure if it should be CW.)

'Universal algebra - roughly - is the study, construed broadly, of classes of algebraic structures (in a given language) defined by equations. Of course, it is really much more than that, but that's the best one-sentence definition I can give.

I'm very new to universal algebra. So far I've found it incredibly interesting, mainly because it looks at things I was already interested in from a new (to me) perspective, and that's always good; but I don't at all have a firm command of even the basics. For example, the recent question Relatively free algebras in a variety generated by a single algebra made me realize that I'd naively accepted a very false statement: until I thought about it, I'd sort of taken for granted that $A$ is always relatively free in $Var(A)$.

I'm sure this isn't the only false belief I have about universal algebra, and I'm sure I'll hold more in the future; and I'm also sure I'm not alone in this. So my question is:

What are some notable counterexamples (to reasonable hypotheses a student in universal algebra might have) in universal algebra?

I'm specifically interested in universal algebra because, well, it's fairly universal; it seems reasonable that a counterexample in universal algebra would be of interest to algebraists of many different stripes, and hopefully many outside algebra as well. At the same time, universal algebra is distinct enough that counterexamples in universal algebra would hopefully have their own flavor not found necessarily in questions like "counterexamples in group theory," "counterexamples in ring theory," etc. In that vein, I'd especially appreciate counterexamples about topics firmly within universal algebra - say, congruence lattices, or Mal'cev conditions - which nonetheless have "something to say" to other areas of mathematics.'

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    $\begingroup$ I am mildly amused at the request of existential examples to aid in studying a subject ostensibly laden with universal statements. I suppose MO could stand another big list question. $\endgroup$
    – The Masked Avenger
    Commented Jan 9, 2014 at 2:02
  • $\begingroup$ This seems like a fine question to me, though I know very little about universal algebra and thus am probably not the best judge. $\endgroup$
    – Andy Putman
    Commented Jan 9, 2014 at 2:20
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    $\begingroup$ If they are still available at Cal, you might get George Bergman's 245 notes. If any one alive can do counterexamples in UA, he could likely do them better. $\endgroup$
    – The Masked Avenger
    Commented Jan 9, 2014 at 2:34
  • $\begingroup$ Posted: mathoverflow.net/questions/154017/…. $\endgroup$
    – Noah Schweber
    Commented Jan 9, 2014 at 2:55
  • $\begingroup$ I have seen that question in the hot list on physics just a minute ago ... :-/ $\endgroup$
    – Dilaton
    Commented Jan 9, 2014 at 6:57

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