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A juggler’s dozen of easy† problems

(† Well, easily formulated . . .)

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Is every dualizable finite algebra of finite signature finitely based? What is the likelihood that a random finite modular lattice directly decomposes into an even number of directly indecomposable lattices? Is the algebra \({\langle \mathbb{N},+, \cdot, {n\atopwithdelims ()k},!,0, 1 \rangle }\) finitely based? Is it decidable, given a finite lattice \({\mathbf{L}}\) and a finite algebra \({\mathbf{A}}\), whether \({\mathbf{L}}\) can be embedded into the congruence lattice of an algebra belonging to the variety generated by \({\mathbf{A}}\)? What is the Nullstellensatz for free lattices? Which finite automatic algebras are dualizable?

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References

  1. Al Dhamri, N.: Dualities for quasi-varieties of bands. Semigroup Forum 88, 417–432 (2014). http://dx.doi.org/10.1007/s00233-013-9541-4

  2. Baker K.A.: Finite equational bases for finite algebras in a congruence-distributive equational class. Advances in Math. 24, 207–243 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baker K.A., McNulty G.F., Werner H.: Shift-automorphism methods for inherently nonfinitely based varieties of algebras. Czechoslovak Math. J. 39(114), 53–69 (1989)

    MathSciNet  Google Scholar 

  4. Baker K.A., Pixley A.F.: Polynomial interpolation and the Chinese remainder theorem for algebraic systems. Math. Z. 143, 165–174 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baker, K.A., Wang, J.: Definable principal subcongruences. Algebra Universalis 47, 145–151 (2002). http://dx.doi.org/10.1007/s00012-002-8180-5

  6. Banaschewski, B.: Projective covers in categories of topological spaces and topological algebras. In: General Topology and its Relations to Modern Analysis and Algebra, III (Proc. Conf., Kanpur, 1968), pp. 63–91. Academia, Prague (1971)

  7. Barto, L., Kozik, M.: Absorbing subalgebras, cyclic terms, and the constraint satisfaction problem. Log. Methods Comput. Sci. 8(1), 1:07, 27 (2012). http://dx.doi.org/10.2168/LMCS-8(1:7)2012

  8. Bentz, W., Davey, B.A., Pitkethly, J.G., Willard, R.: Dualizability of automatic algebras. J. Pure Appl. Algebra 218, 1324–1345 (2014) http://dx.doi.org/10.1016/j.jpaa.2013.11.020

  9. Birkhoff, G.: Universal algebra. In: Proc. First Canadian Math. Congress, Montreal, 1945, pp. 310–326. University of Toronto Press, Toronto (1946)

  10. Birkhoff, G.: Lattice Theory. American Mathematical Society Colloquium Publications, vol. 25, revised edition. American Mathematical Society, New York (1948)

  11. Boozer IV, J.F.: On the finite axiomatizability of equational theories of automatic algebras. ProQuest LLC, Ann Arbor, MI. PhD thesis, University of South Carolina (2010) http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:3413199.

  12. Bryant, R.M.: The laws of finite pointed groups. Bull. London Math. Soc. 14, 119–123 (1982). http://dx.doi.org/10.1112/blms/14.2.119

  13. Burris, S.N.: Number Theoretic Density and Logical Limit Laws. Mathematical Surveys and Monographs, vol. 86. American Mathematical Society, Providence (2001)

  14. Cacioppo, R.: Non-finitely based pseudovarieties and inherently non-finitely based varieties. Semigroup Forum 47, 223–226 (1993). http://dx.doi.org/10.1007/BF02573759

  15. Clark, D.M., Davey, B.A.: Natural Dualities for the Working Algebraist. Cambridge Studies in Advanced Mathematics vol. 57. Cambridge University Press, Cambridge (1998)

  16. Clark, D.M., Davey, B.A., Pitkethly, J.G., Rifqui, D.L.: Flat unars: the primal, the semi-primal and the dualisable. Algebra Universalis 63, 303–329 (2010). http://dx.doi.org/10.1007/s00012-010-0080-5.

  17. Clark, D.M., Idziak, P.M., Sabourin, L.R., Szabó, C., Willard, R.: Natural dualities for quasivarieties generated by a finite commutative ring. Algebra Universalis 46, 285–320 (2001) (the Viktor Aleksandrovich Gorbunov memorial issue). http://dx.doi.org/10.1007/PL00000344.

  18. Davey, B.A.: Duality theory on ten dollars a day. In: Algebras and Orders (Montreal, PQ, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 389, pp. 71–111. Kluwer Acad. Publ., Dordrecht (1993)

  19. Davey, B.A., Heindorf, L., McKenzie, R.: Near unanimity: an obstacle to general duality theory. Algebra Universalis 33, 428–439 (1995). http://dx.doi.org/10.1007/BF01190710

  20. Davey, B.A., Idziak, P., Lampe, W.A., McNulty, G.F.: Dualizability and graph algebras. Discrete Math. 214, 145–172 (2000). http://dx.doi.org/10.1016/S0012-365X(99)00225-3

  21. Davey, B.A., Jackson, M., Pitkethly, J.G., Talukder, M.R.: Natural dualities for semilattice-based algebras. Algebra Universalis 57, 463–490 (2007). http://dx.doi.org/10.1007/s00012-007-2061-x

  22. Davey, B.A., Knox, B.J.: From rectangular bands to k-primal algebras. Semigroup Forum 64, 29–54 (2002). http://dx.doi.org/10.1007/s002330010023

  23. DeMeo, W.: Interval enforceable properties of finite groups. Communications in Algebra (to appear)

  24. DeMeo, W.: Expansions of finite algebras and their congruence lattices. Algebra Universalis 69, 257–278 (2013). http://dx.doi.org/10.1007/s00012-013-0226-3

  25. DeMeo, W.J.: Congruence Lattices of Finite Algebras. ProQuest LLC, Ann Arbor, MI. PhD thesis, University of Hawai’i at Manoa (2012). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:3520678.

  26. van den Dries, L.: A completeness theorem for trigonometric identities and various results on exponential functions. Proc. Amer. Math. Soc. 96, 345–352 (1986). http://dx.doi.org/10.2307/2046179

  27. Dziobiak, W.: On infinite subdirectly irreducible algebras in locally finite equational classes. Algebra Universalis 13, 393–394 (1981). http://dx.doi.org/10.1007/BF02483850

  28. Eilenberg S., Schützenberger M.P.: On pseudovarieties. Advances in Math. 19, 413–418 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. Freese, R.: On the two kinds of probability in algebra. Algebra Universalis 27, 70–79 (1990). http://dx.doi.org/10.1007/BF01190254

  30. Freese, R., Ježek, J., Nation, J.B.: Free lattices, Mathematical Surveys and Monographs, vol. 42. American Mathematical Society, Providence (1995)

  31. Freese, R., McKenzie, R.: Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc. 264, 419–430 (1981). http://dx.doi.org/10.2307/1998548

  32. Grätzer G., Schmidt E.T.: Characterizations of congruence lattices of abstract algebras. Acta Sci. Math. (Szeged) 24, 34–59 (1963)

    MathSciNet  MATH  Google Scholar 

  33. Gurevi c, R.: Equational theory of positive numbers with exponentiation is not finitely axiomatizable. Ann. Pure Appl. Logic 49, 1–30 (1990). http://dx.doi.org/10.1016/0168-0072(90)90049-8

  34. Henson, C.W., Rubel, L.A.: Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions. Trans. Amer. Math. Soc. 282, 1–32 (1984). http://dx.doi.org/10.2307/1999575

  35. Henson, C.W., Rubel, L.A.: Correction to: “Some applications of Nevanlinna theory to mathematical logic: identities of exponential functions” [Trans. Amer. Math. Soc. 282, 1–32 (1984)]. Trans. Amer. Math. Soc. 294, 381 (1986). http://dx.doi.org/10.2307/2000139

  36. Hobby, D.: Nondualisable semigroups. Bull. Austral. Math. Soc. 65, 491–502 (2002). http://dx.doi.org/10.1017/S0004972700020542

  37. Hobby, D., McKenzie, R.: The structure of finite algebras, Contemporary Mathematics, vol. 76. American Mathematical Society, Providence (1988)

  38. Hofmann, K.H., Mislove, M., Stralka, A.: The Pontryagin duality of compact O-dimensional semilattices and its applications. Lecture Notes in Mathematics, vol. 396. Springer, Berlin (1974)

  39. Isaev, I.M.: Inherently non-finitely based varieties of algebras. Sibirsk. Mat. Zh. 30, 75–77 (1989). http://dx.doi.org/10.1007/BF00970911

  40. Jackson, M.: Dualisability of finite semigroups. Internat. J. Algebra Comput. 13, 481–497 (2003). http://dx.doi.org/10.1142/S0218196703001535

  41. Ježek J.: Intervals in the lattice of varieties. Algebra Universalis 6, 147–158 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ježek, J.: The lattice of equational theories. I. Modular elements. Czechoslovak Math. J. 31(106), 127–152 (1981) (with a loose Russian summary)

  43. Ježek, J.: The lattice of equational theories. II. The lattice of full sets of terms. Czechoslovak Math. J. 31(106), 573–603 (1981)

  44. Ježek, J.: The lattice of equational theories. III. Definability and automorphisms. Czechoslovak Math. J. 32(107), 129–164 (1982)

  45. Ježek, J.: The lattice of equational theories. IV. Equational theories of finite algebras. Czechoslovak Math. J. 36(111), 331–341 (1986)

  46. Jónsson, B.: The unique factorization problem for finite relational structures. Colloq. Math. 14, 1–32 (1966)

  47. Kearnes, K.A., Kiss, E.W.: The shape of congruence lattices. Mem. Amer. Math. Soc. 222(1046), viii+169 (2013). http://dx.doi.org/10.1090/S0065-9266-2012-00667-8

  48. Kearnes, K., Szendrei, Á., Willard, R.: A finite basis theorem for difference-term varieties with a finite residual bound. Trans. Amer. Math. Soc. (to appear)

  49. Kearnes, K.A., Willard, R.: Residually finite, congruence meet-semidistributive varieties of finite type have a finite residual bound. Proc. Amer. Math. Soc. 127, 2841–2850 (1999). http://dx.doi.org/10.1090/S0002-9939-99-05097-2

  50. Lampe, W.A.: A property of the lattice of equational theories. Algebra Universalis 23, 61–69 (1986). http://dx.doi.org/10.1007/BF01190912

  51. Lampe, W.A.: A perspective on algebraic representations of lattices. Algebra Universalis 31, 337–364 (1994). http://dx.doi.org/10.1007/BF01221791

  52. Lampe, W.A.: Congruences, equational theories and lattice representations. Period. Math. Hungar. 32, 65–75 (1996). http://dx.doi.org/10.1007/BF01879733

  53. Lampe, W.A.: Results and problems on congruence lattice representations. Algebra Universalis 55, 127–135 (2006). http://dx.doi.org/10.1007/s00012-006-1990-0 (Special issue dedicated to Walter Taylor)

  54. Lampe, W.A., McNulty, G.F., Willard, R.: Full duality among graph algebras and flat graph algebras. Algebra Universalis 45, 311–334 (2001) (Conference on Lattices and Universal Algebra, Szeged 1998)

  55. Lyndon R.C.: Identities in finite algebras. Proc. Amer. Math. Soc. 5, 8–9 (1954)

    Article  MathSciNet  MATH  Google Scholar 

  56. Mal′cev, A.I.: Some borderline problems of algebra and logic. In: Proc. Internat. Congr. Math. (Moscow, 1966), pp. 217–231. Izdat. “Mir”, Moscow (1968)

  57. Maróti, M., McKenzie, R.: Existence theorems for weakly symmetric operations. Algebra Universalis 59, 463–489 (2008). http://dx.doi.org/10.1007/s00012-008-2122-9

  58. Martin, C.: Equational theories of natural numbers and transfinite ordinals. PhD thesis, University of California, Berkeley (1973)

  59. McKenzie, R.: Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties. Algebra Universalis 8, 336–348 (1978)

  60. McKenzie, R.: Finite equational bases for congruence modular varieties. Algebra Universalis 24, 224–250 (1987). http://dx.doi.org/10.1007/BF01195263

  61. McKenzie, R.: The residual bounds of finite algebras. Internat. J. Algebra Comput. 6, 1–28 (1996). http://dx.doi.org/10.1142/S0218196796000027

  62. McKenzie, R.: The residual bound of a finite algebra is not computable. Internat. J. Algebra Comput. 6, 29–48 (1996). http://dx.doi.org/10.1142/S0218196796000039

  63. McKenzie, R.: Tarski’s finite basis problem is undecidable. Internat. J. Algebra Comput. 6, 49–104 (1996). http://dx.doi.org/10.1142/S0218196796000040

  64. McKenzie, R.N., McNulty, G.F., Taylor, W.F.: Algebras, Lattices, Varieties, vol. I. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA (1987)

  65. McNulty, G.F., Székely, Z., Willard, R.: Equational complexity of the finite algebra membership problem. Internat. J. Algebra Comput. 18, 1283–1319 (2008). http://dx.doi.org/10.1142/S0218196708004913

  66. Murskiĭ, V.L.: The number of k-element algebras with a binary operation which do not have a finite basis of identities. Problemy Kibernet. (35), 5–27, 208 (1979)

  67. Nickodemus, M.H.: Natural dualities for finite groups with abelian Sylow subgroups. ProQuest LLC, Ann Arbor, MI. PhD thesis, University of Colorado at Boulder (2007)

  68. Nurakunov, A.M.: Unreasonable lattices of quasivarieties. Internat. J. Algebra Comput. 22, 125006, 17pp. (2012). http://dx.doi.org/10.1142/S0218196711006728

  69. Owens, K.S.: On Inherently Nonfinitely Based Varieties. PhD thesis, University of South Carolina (2009)

  70. Owens, K.S.: Every shift automorhism variety has an infinite subdirectly irreducible member. J. Austral. Math. Soc. 88, 231–238 (2010). http://dx.doi.org/10.1017/S1446788709000317

  71. Pálfy, P.P., Pudlák, P.: Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups. Algebra Universalis 11, 22–27 (1980). http://dx.doi.org/10.1007/BF02483080

  72. Park, R.E.: Equational classes on non-associative ordered algebras. PhD thesis, University of California, Los Angeles (1976)

  73. Perkins, P.: Basic questions for general algebras. Algebra Universalis 19, 16–23 (1984). http://dx.doi.org/10.1007/BF01191487

  74. Petkovšek, M., Wilf, H.S., Zeilberger, D.: A =  B. A K Peters, Wellesley, (1996) (with a foreword by Donald E. Knuth)

  75. Pitkethly, J., Davey, B.: Dualisability, Unary algebras and Beyond. Advances in Mathematics, vol. 9. Springer, New York (2005)

  76. Pitkethly, J.G.: Uncountably many dualisable algebras. Internat. J. Algebra Comput. 21, 825–839 (2011). http://dx.doi.org/10.1142/S0218196711006534

  77. Pontryagin, L.S.: Topological Groups. Translated from the second Russian edition by Arlen Brown. Gordon and Breach Science Publishers, New York (1966)

  78. Priestley, H.A.: Ordered topological spaces and the representation of distributive lattices. Proc. London Math. Soc. (3) 24, 507–530 (1972)

  79. Pudlák, P., Tůma, J.: Every finite lattice can be embedded in a finite partition lattice. Algebra Universalis 10, 74–95 (1980). http://dx.doi.org/10.1007/BF02482893

  80. Quackenbush, R.W.: Equational classes generated by finite algebras. Algebra Universalis 1, 265–266 (1971/72)

  81. Quackenbush, R., Szabó, C.: Nilpotent groups are not dualizable. J. Austral. Math. Soc. 72, 173–179 (2002). http://dx.doi.org/10.1017/S1446788700003827

  82. Quackenbush, R., Szabó, C.: Strong duality for metacyclic groups. J. Aust. Math. Soc. 73, 377–392 (2002). http://dx.doi.org/10.1017/S1446788700009034

  83. Sapir, M.V.: Inherently non-finitely based finite semigroups. Mat. Sb. (N.S.) 133(175)(2), 154–166, 270 (1987)

  84. Sapir, M.V.: Problems of Burnside type and the finite basis property in varieties of semigroups. Izv. Akad. Nauk SSSR Ser. Mat. 51(2), 319–340, 447 (1987)

  85. Sapir M.V.: Sur la propriété de base finie pour les pseudovariétés de semigroupes finis. C. R. Acad. Sci. Paris Sér. I Math. 306(20), 795–797 (1988)

    MathSciNet  MATH  Google Scholar 

  86. Siggers, M.H.: A strong Mal’cev condition for locally finite varieties omitting the unary type. Algebra Universalis 64, 15–20 (2010). http://dx.doi.org/10.1007/s00012-010-0082-3

  87. Stone, M.H.: The theory of representations for Boolean algebras. Trans. Amer. Math. Soc. 40, 37–111 (1936). http://dx.doi.org/10.1090/S0002-9947-1936-1501865-8

  88. Székely, Z.: Complexity of the Finite Algebra Membership Problem for Varieties. PhD thesis, University of South Carolina, Columbia (1998)

  89. Taylor W.: Varieties obeying homotopy laws. Canad. J. Math. 29, 498–527 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  90. Whitman P.M.: Lattices, equivalence relations, and subgroups. Bull. Amer. Math. Soc. 52, 507–522 (1946)

    Article  MathSciNet  MATH  Google Scholar 

  91. Willard, R.: A finite basis theorem for residually finite, congruence meet-semidistributive varieties. J. Symbolic Logic 65, 187–200 (2000). http://dx.doi.org/10.2307/2586531

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    George F. McNulty

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McNulty, G.F. A juggler’s dozen of easy† problems. Algebra Univers. 74, 17–34 (2015). https://doi.org/10.1007/s00012-015-0333-4

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