On lattices connected with various types of classes of algebraic structures
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 2, pp. 167-179

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This survey paper reviews some recent results related to various derived lattices connected with various types of classes of algebraic structures which were obtained by the authors.
Keywords: axiomatizable class, variety, quasivariety, prevariety, finitary prevariety, identity, quasi-identity, lattice, subsemilattice lattice.
A. Nurakunov; M. Semenova; A. Zamojska-Dzienio. On lattices connected with various types of classes of algebraic structures. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 154 (2012) no. 2, pp. 167-179. http://geodesic.mathdoc.fr/item/UZKU_2012_154_2_a16/
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[1] Birkhoff G., “Universal algebra”, Proc. First Can. Math. Congress (Montreal, 1945), Univ. of Toronto Press, Toronto, 1946, 310–326 | MR | Zbl

[2] Maltsev A. I., “Some Borderline problems of algebra and logic”, Proc. Int. Congr. Math. (Moscow, 1966), Mir, Moscow, 1968, 217–231 | MR

[3] Adaricheva K. V., Gorbunov V. A., Dziobiak W., “Algebraic atomistic lattices of quasivarieties”, Algebra Logic, 36:4 (1997), 213–225 | MR | Zbl

[4] Gorbunov V. A., Algebraic Theory of Quasivarieties, Plenum, N.Y., 1998, 316 pp. | MR

[5] Adaricheva K., Nation J. B., Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part I, 2012, 27 pp., arXiv: 1106.2203v3[math.RA] | MR

[6] Adaricheva K., Nation J. B., Lattices of quasi-equational theories as congruence lattices of semilattices with operators, Part II, 2012, 19 pp., arXiv: 1106.2204v3[math.RA] | MR

[7] Nation J. B., Lattices of theories in languages without equality, URL: , 2010, 15 pp. http://www.math.hawaii.edu/~jb/coopers.pdf

[8] McNulty G. F., “How Undecidable is the Elementary Theory of the Lattice of Equational Theories?”, Int. Conf. on Algebras and Lattices (June, 2010, Prague, Czech Republic) URL: http://www.karlin.mff.cuni.cz/~ical/presentations/mcnulty.pdf

[9] Adams M., Adaricheva K., Dziobiak W., Kravchenko A., “Open questions related to the problem of Birkhoff and Maltsev”, Studia Logica, 78:1–2 (2004), 357–378 | DOI | MR | Zbl

[10] Agliano P., Nation J. B., “Lattices of pseudovarieties”, J. Austral. Math. Soc. Ser. A, 46:3 (1989), 177–183 | DOI | MR | Zbl

[11] Nurakunov A. M., “Quasivariety lattices having no reasonable description”, Int. J. Algebra Comput., 22:3 (2012), 1–17 | DOI | MR

[12] Pal'chunov D. E., “Lattices of relatively axiomatizable classes”, Formal Concept Analysis, 5th Int. Conf., ICFCA 2007, Proc., Lecture Notes in Artificial Intelligence, 4390, Springer-Verlag, Berlin–Heidelberg, 2007, 221–239

[13] Gorbunov V. A., “Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. II”, Algebra Logic, 34:4 (1995), 203–218 | DOI | MR | Zbl

[14] Gorbunov V. A., Tumanov V. I., “The structure of lattices of quasivarieties”, Proc. Inst. Math., Sib. Branch Acad. Sci. USSR, 2, 1982, 12–44 (in Russian) | MR | Zbl

[15] Vernitski A., “Finite quasivarieties and self-referential conditions”, Studia Logica, 78 (2004), 337–348 | DOI | MR | Zbl

[16] Banaschewski B., Herrlich H., “Subcategories defined by implications”, Houston J. Math., 2 (1976), 149–171 | MR | Zbl

[17] Semenova M. V., Zamojska-Dzienio A., “On lattices of subclasses”, Sib. J. Math., 53:5 (2012), 1111–1132 URL: http://www.mini.pw.edu.pl/~azamojsk/LpSZD.pdf | DOI | MR | Zbl

[18] Gorbunov V. A., Tumanov V. I., “A class of lattices of quasivarieties”, Algebra Logic, 19 (1980), 38–52 | DOI | MR | Zbl

[19] Gorbunov V. A., Tumanov V. I., “On the structure of lattices of quasivarieties”, Sov. Math. Dokl., 22 (1980), 333–336 | MR | Zbl

[20] Nurakunov A. M., “Finite lattices as relative congruence lattices of finite algebras”, Algebra Universalis, 57:2 (2007), 207–214 | DOI | MR | Zbl

[21] Nurakunov A. M., “Equational theories as congruences of enriched monoids”, Algebra Universalis, 58:3 (2008), 357–372 | DOI | MR | Zbl

[22] Gorbunov V. A., “Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. I”, Algebra Logic, 34:2 (1995), 73–86 | DOI | MR | Zbl

[23] Gorbunov V. A., “Structure of lattices of varieties and lattices of quasivarieties: Similarity and difference. III”, Algebra Logic, 34:6 (1995), 359–370 | DOI | MR | Zbl

[24] Rubin H., Rubin J. E., Equivalents of the Axiom of Choice, v. II, Studies in Logic and the Foundations of Mathematics, 116, North-Holland, Elsevier Science, Amsterdam–N.Y.–Oxford, 1985, 322 pp. | MR | Zbl

[25] Lampe W. A., “A perspective on algebraic representations of lattices”, Algebra Universalis, 31 (1994), 337–364 | DOI | MR | Zbl

[26] Freese R., Ježek J., Nation J. B., Free lattices, Mathematical Surveys and Monographs, 42, Am. Math. Soc., Providence, RI, 1995, 293 pp. | DOI | MR | Zbl

[27] Repnitskiǐ V. B., “On finite lattices which are embeddable in subsemigroup lattices”, Semigroup Forum, 46 (1993), 388–397 | DOI | MR | Zbl