Research monographs and open problems in universal algebra

Question

I am someone who is very interested in the mathematical subfield of universal algebra. I want to know, what are some significant open problems in universal algebra? I would like a list of such problems. My second related question is, what are some research monographs (that is, written for the expert) on universal algebra? I have read some introductory books on universal algebra, but not any research monographs. Perhaps the research monographs would also list some open problems.

Answer 1

The question has been asked and answered, but let me add more information to amrsa's answer.

Research Monographs. I feel that anyone who wants to read a Research Monograph in general algebra should start with

1. Freese, Ralph; McKenzie, Ralph
   Commutator theory for congruence modular varieties.
   London Mathematical Society Lecture Note Series, 125. Cambridge University Press, Cambridge, 1987.

It is not hard to read. Some fluency in this topic is indispensable for more advanced work.

For the rest of this list I restrict my attention to Research Monographs written in English during the last 40 years. Even with these restrictions, I have restrict myself further to those I know best. These are listed in reverse chronological order

2. Adaricheva, Kira; Hyndman, Jennifer; Nation, J. B.; Nishida, Joy N.
   A primer of subquasivariety lattices.
   CMS/CAIMS Books in Mathematics, 3. Springer, Cham, [2022], ©2022.

3. Bodirsky, Manuel
   Complexity of infinite-domain constraint satisfaction.
   Lecture Notes in Logic, 52. Cambridge University Press, Cambridge; Association for Symbolic Logic, Ithaca, NY, 2021.

4. Kearnes, Keith A.; Kiss, Emil W.
   The shape of congruence lattices.
   Mem. Amer. Math. Soc. 222, no. 1046, 2013.

5. Pitkethly, Jane; Davey, Brian
   Dualisability. Unary algebras and beyond.
   Advances in Mathematics (Springer), 9. Springer, New York, 2005.

6. Berman, Joel; Idziak, Paweł M.
   Generative complexity in algebra.
   Mem. Amer. Math. Soc. 175, no. 828 (2005).

7. Romanowska, Anna B.; Smith, Jonathan D. H.
   Modes.
   World Scientific Publishing Co., Inc., River Edge, NJ, 2002.

8. Gorbunov, Viktor A.
   Algebraic theory of quasivarieties.
   Translated from the Russian. Siberian School of Algebra and Logic. Consultants Bureau, New York, 1998.

9. McKenzie, Ralph; Valeriote, Matthew
   The structure of decidable locally finite varieties.
   Progress in Mathematics, 79. Birkhäuser Boston, Inc., Boston, MA, 1989.

10. Hobby, David; McKenzie, Ralph
    The structure of finite algebras.
    Contemporary Mathematics, 76. American Mathematical Society, Providence, RI, 1988.

11. Szendrei, Ágnes
    Clones in universal algebra.
    Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], 99. Presses de l'Université de Montréal, Montreal, QC, 1986.

12. García, O. C.; Taylor, W.
    The lattice of interpretability types of varieties.
    Mem. Amer. Math. Soc. 50, no. 305 (1984).

13. Gumm, H. Peter
    Geometrical methods in congruence modular algebras.
    Mem. Amer. Math. Soc. 45, no. 286 (1983).

Problem Lists. Miklós Maróti has been collecting lists of open problems in universal algebra since 1998. You can see them at his web page. There are 11 lists there. (Some of the problems on these lists have been solved.)

Answer 2

From the top of my head, I could think of these:

1. Commutator Theory for Congruence Modular Varieties by Ralph Freese and Ralph McKenzie (freely available here).

2. The Structure of Finite Algebras by David Hobby and Ralph McKenzie (freely available here).

3. An Easy Way to Minimal Algebras by Emil Kiss (freely available here)

4. The Shape of Congruence Lattices by Keith Kearnes and Emil Kiss (freely available here).

5. Universal Algebra by Jaroslav Jezek (freely available here); this one I think is mostly introductory, but has some material not found in other introductory texts on the subject (?); I only read a few results from it, so I can't say for sure.

These are probably not among the most up to date, but are good references.
Surely others will think of other resources.

Notice also that the expression "freely available here" just means that I could find a link to the document without having to make any registration.
I'm not acquainted with the subject of rights of access (but obviously copyrights apply, but these are not an issue here).

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Edit. I'm not sure it falls into this category, also because I don't know what exact subjects are covered, but it seems that the series Algebras, Lattices, Varieties, by McKenzie, McNulty and Taylor, whose first volume is from 1987, was finally updated (and completed?) with Volume II and Volume III last year.
(These volumes with also the participation of Freese.)