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Characterizing Mal’cev conditions

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References

  1. J. AczélProof of a theorem on distributive type hyperidentities, Algebra Universalis1 (1971), 1–7.

    MATH  MathSciNet  Google Scholar 

  2. A. A. Akataev and D. M. Smirnov,The lattice of submanifolds of an algebraic manifold (in Russian), Algebra i Logika7, (1968), 5–25; English translation: Algebra and Logic7 (1968), 3–13.

    MathSciNet  Google Scholar 

  3. A. K. Austin,A note on models of identities, Proc. Amer. Math. Soc.16 (1965), 522–523.

    Article  MATH  MathSciNet  Google Scholar 

  4. A. K. Austin,Finite models for laws in two variables, Proc. Amer. Math. Soc.17 (1966), 1410–1412.

    Article  MATH  MathSciNet  Google Scholar 

  5. J. L. Bell and A. B. Slomson,Models and ultraproducts: an, introduction, North-Holland, Amsterdam, 1969.

    MATH  Google Scholar 

  6. G. Birkhoff,On the structure of abstract algebras, Proc. Cambridge Philos. Soc.31 (1935), 433–454.

    Google Scholar 

  7. G. Birkhoff,Lattice theory (3rd edition), Amer. Math. Soc. Colloq. Publ. #25, Providence, 1967.

    MATH  Google Scholar 

  8. G. Birkhoff, and O. Frink,Representation of lattices by sets. Trans. Amer. Math. Soc.64 (1948), 299–316.

    Article  MATH  MathSciNet  Google Scholar 

  9. G. Birkhoff and J. D. Lipson,Heterogeneous algebras, J. Combinatorial Theory8 (1970), 115–133.

    Article  MATH  MathSciNet  Google Scholar 

  10. S. Burris,Models in equational theories of unary algebras, Algebra Universalis1 (1972), 386–392.

    MATH  MathSciNet  Google Scholar 

  11. A. Church,Introduction to mathematical logic I, Princeton University Press, Princeton, 1956.

    MATH  Google Scholar 

  12. P. M. Cohn,Universal algebra, Harper and Row, New York, 1965.

    MATH  Google Scholar 

  13. B. Csákány,On the equivalence of certain classes of algebraic systems (in Russian), Acta Sci. Math. (Szeged)23 (1962), 46–57.

    MathSciNet  Google Scholar 

  14. B. Csákány,Primitive classes of algebras which are equivalent to classes of semi-modules and modules (in Russian) Acta Sci. Math. (Szeged)24 (1963), 157–164.

    MATH  MathSciNet  Google Scholar 

  15. B. Csákány,Abelian properties of primitive classes of universal algebras (in Russian), Acta Sci. Math. (Szeged)25, (1964), 202–208.

    MATH  MathSciNet  Google Scholar 

  16. B. Csákány,Characterization of regular varieties, Acta Sci. Math. (Szeged)31 (1970), 187–189.

    MATH  MathSciNet  Google Scholar 

  17. A. Day,A characterization of modularity for congruence lattices of algebras, Canad. Math. Bull.12 (1969), 167–173.

    MATH  MathSciNet  Google Scholar 

  18. A. Day,A note on the congruence extension property, Algebra Universalis1 (1971), 234–235.

    MATH  MathSciNet  Google Scholar 

  19. T. Evans,Properties of algebras almost equivalent to, identities, J. London Math. Soc.37 (1962), 53–59.

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Evans,The spectrum of a variety, Z. Math. Logik Grundlagen Math.,13 (1967), 213–218.

    Article  MATH  MathSciNet  Google Scholar 

  21. S. Fajtlowicz,Birkhoff’s theorem in the category of non-indexed algebras, Bull. Acad. Pol. Sci., Sér. sci. math. astr. et phys.17 (1969), 273–275.

    MATH  MathSciNet  Google Scholar 

  22. S. Fajtlowicz,Algebras of homomorphisms, Rend. Math. (6)3 (1970), 523–527.

    MATH  MathSciNet  Google Scholar 

  23. S. Fajtlowicz, W. Holsztyński, J. Mycielski, and B. Weglorz,On powers of bases in some compact algebras, Colloq. Math.19 (1968), 43–46.

    MATH  MathSciNet  Google Scholar 

  24. W. Felscher, Equational maps, 121–161 in H. A. Schmidt, K. Schütte and H.-J. Thiele (eds.)Contributions to mathematical logic, North-Holland, Amsterdam, 1968.

    Google Scholar 

  25. K. Fichtner,Varieties of universal algebras with ideals (in Russian), Mat. Sbornik,75 (117) (1968), 445–453; English translation: Math. USSR-Sbornik4 (1968), 411–418.

    MathSciNet  Google Scholar 

  26. K. Fichtner,Distributivity and modularity in varieties of algebras, Acta Sci. Math. (Szeged)33 (1972), 343–348.

    MATH  MathSciNet  Google Scholar 

  27. G. A. Fraser and A. Horn,Congruence relations in direct products, Proc. Amer. Math. Soc.26 (1970), 390–394.

    Article  MATH  MathSciNet  Google Scholar 

  28. P. Freyd,Abelian categories, Harper and Row, New York, 1964.

    MATH  Google Scholar 

  29. N. Funayama and T. Nakayama,On the distributivity of a lattice of lattice-congruences, Proc. Imp. Acad. Tokyo18 (1942), 553–554.

    MATH  MathSciNet  Google Scholar 

  30. A. Goetz,On weak isomorphisms and weak homomorphisms of abstract algebras, Colloq. Math.14 (1966), 163–167.

    MATH  MathSciNet  Google Scholar 

  31. A. Goetz,A generalization of the notion of direct product of universal algebras, Colloq. Math.22 (1971), 167–176.

    MATH  MathSciNet  Google Scholar 

  32. A. Goetz and C. Ryll-Nardzewski,On bases of abstract algebras, Bull. Acad. Pol. Sci., Sér. sci. math., astr. et phys.8 1960), 157–161.

    MATH  MathSciNet  Google Scholar 

  33. G. Grätzer,On spectra of classes of universal algebras, Proc. Amer. Math. Soc.18 (1967), 729–735.

    Article  MATH  MathSciNet  Google Scholar 

  34. G. Grätzer,Universal algebra, van Nostrand, Princeton, 1968.

    MATH  Google Scholar 

  35. G. Grätzer,Two Mal’cev-type theorems in universal algebra, J. Combinatorial theory8 (1970), 334–342.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Grätzer, H. Lakser, and J. Płonka,Joins and direct products of equational classes, Canad. Math. Bull.12 (1969), 741–744.

    MATH  MathSciNet  Google Scholar 

  37. J. Hagemann,On regular and weakly regular congruences, Algebra Universalis (to appear).

  38. J. Hagemann and A. Mitschke,On n-permutable congruences Algebra Universalis (to appear).

  39. L. Henkin, J. D. Monk, and A. Tarski,Cylindric algebras, North-Holland, Amsterdam, 1971.

    MATH  Google Scholar 

  40. P. J. HigginsAlgebras with a scheme of operators, Math. Nachr.27 (1963), 115–132.

    Article  MATH  MathSciNet  Google Scholar 

  41. D. Hilbert,The foundations of geometry, translated by E. J. Townsend, Open Court, Chicago, 1902, 1910.

    MATH  Google Scholar 

  42. T.-K. Hu,On equational classes of algebras in which congruences of finite direct products are induced by congruences on their factors, mimeographed, Southern Illinois University (1968?).

  43. T.-K. Hu,Stone duality for primal algebra theory, Math. Z.110 (1969), 180–198.

    Article  MATH  MathSciNet  Google Scholar 

  44. T.-K. Hu,Direct factorization in certain classes of universal algebras, Abstract 672–304, Notices Amer. Math. Soc.17 (1970), 169.

    Google Scholar 

  45. T.-K. Hu,On the topological duality for primal algebra theory, Algebra Universalis1 (1971), 152–154.

    MATH  MathSciNet  Google Scholar 

  46. T.-K. Hu and P. Kelenson,Independence and direct factorization of universal algebras, Math. Nachr.51 (1971), 83–99.

    Article  MATH  MathSciNet  Google Scholar 

  47. B. Jónsson,Algebras whose congruence lattices are distributive, Math. Scand.21 (1967), 110–121.

    MATH  MathSciNet  Google Scholar 

  48. L. Kalmár,Contributions to the reduction theory of the decision problem, Fourth paper:Reduction to a finite set of individuals Acta Math. Acad. Sci. Hung.2 (1951), 125–142.

    Article  MATH  Google Scholar 

  49. D. KellyBasic equations: word problems and Mal’cev conditions, Abstract 701-08-4, Notices Amer. Math. Soc.20 (1973), A-54.

    Google Scholar 

  50. G. Kreisel and J. L. Krivine,Elements of mathematical logic (model theory), North-Holland, Amsterdam, 1967.

    MATH  Google Scholar 

  51. J.-L. Lagrange,Réflexions sur la résolution algébrique des équations, Nouveaux Mémoires de I’Académie Royale des Sciences et Belles Lettres de Berlin, 1770–1771.

  52. F. W. Lawvere,Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci.50 (1963), 869–872.

    Article  MATH  MathSciNet  Google Scholar 

  53. F. W. Lawvere,Some algebraic problems in the context of functorial semantics of algebraic theories, pp. 51–61 in Lecture Notes in Mathematics61 Springer-Verlag, Berlin, 1968.

    Google Scholar 

  54. F. E. J. Linton,Some aspects of equational, categories, pp. 84–89 in: S. Eilenberg, D. K. Harrison, S. MacLane and H. Röhrl, eds.. Proceedings of the conference on categorical algebra, La Jolla, 1965. Springer-Verlag, Berlin and New York, 1966.

    Google Scholar 

  55. J. Łoś,Direct sums in general algebra, Colloq. Math.14 (1966), 33–38.

    MATH  MathSciNet  Google Scholar 

  56. S., MacLane,Categorical algebra, Bull. Amer. Math. Soc.71 (1965), 40–106.

    Article  MATH  MathSciNet  Google Scholar 

  57. A. I. Mal’cev,On the general theory of algebraic systems (in Russian). Math. Sbornik (N. S.)35 (77) (1954), 3–20. English translation: Amer. Math Soc. Translations (2)27 (1963), 125–142.

    MathSciNet  Google Scholar 

  58. A. I. Mal’cev,The structural characterization of certain classes of algebras (in Russian), Dokl. Akad. Nauk S.S.S.R.120 (1958), 29–32. English translation: Chapter 9 of [60].

    MathSciNet  Google Scholar 

  59. A. I. Mal’cev,Model correspondences, Izv. Akad. Nauk S.S.R. Ser. Mat.23 (1959), 313–316. English translation: Chapter 11 of [60].

    MathSciNet  Google Scholar 

  60. A. I. Mal’cev,The metamathematics of algebraic systems, collected papers 1936–67, translated and edited by B.F. Wells III, North-Holland, Amsterdam, 1971.

    Google Scholar 

  61. E. Marczewski,Independence in abstract algebras-results and problems Colloq. Math.14 (1966), 169–188.

    MATH  MathSciNet  Google Scholar 

  62. G. F. McNulty,The decision problem for equational bases of algebras, Ph.D. Thesis, University of California, Berkeley, 1972.

    Google Scholar 

  63. A. Mitschke,Implication algebras are 3-permutable and 3-distributive, Algebra Universalis1 (1971), 182–186.

    MATH  MathSciNet  Google Scholar 

  64. J. B. Nation,Varieties whose congruences satisfy certain lattice identities, Algebra Universalis (to appear).

  65. B. H. Neumann,On a problem of G. Grätzer, Publ. Math. Debrecen.,14 (1967), 325–329.

    MATH  MathSciNet  Google Scholar 

  66. B. H. Neumann and E. C. Wiegold,A semigroup representation of varieties of algebras, Colloq. Math.14 (1966), 111–114.

    MATH  MathSciNet  Google Scholar 

  67. W. D. Neumann,Representing varieties of algebras by algebras, J. Austral. Math. Soc.11 (1970), 1–8.

    MATH  MathSciNet  Google Scholar 

  68. R. Padmanabhan,Characterization of a class of groupoids, Algebra Universalis1 (1972), 374–382.

    MATH  MathSciNet  Google Scholar 

  69. B. Pareigis,Categories and functors, Academic Press, New York, 1970.

    MATH  Google Scholar 

  70. P. Perkins,Unsolvable problems for equational theories, Notre Dame J. Formal Logic3 (1967), 175–185.

    Article  MathSciNet  Google Scholar 

  71. D. Pigozzi,Amalgamation, congruence-extension and interpolation properties in algebras, Algebra Universalis1 (1972), 269–349.

    MATH  MathSciNet  Google Scholar 

  72. A. F. Pixley,Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc.14 (1963), 105–109.

    Article  MATH  MathSciNet  Google Scholar 

  73. A. F. Pixley,Local Mal’cev conditions, Canad. Math. Bull.15 (1972), 559–568.

    MATH  MathSciNet  Google Scholar 

  74. R. W. Quackenbush,Demi-semi-primal algebras and Mal’cev-type conditions, Math. Z.122 (1971), 166–176.

    Article  MathSciNet  Google Scholar 

  75. R. W. Quackenbush,Models for Mal’cev conditions: n-permutability, Abstract 71T-A182, Notices Amer. Math. Soc.18 (1971), 806.

    Google Scholar 

  76. P. C. Rosenbloom,Post algebras. I. Postulates and general theory, Amer. J. Math.64 (1942), 167–188.

    Article  MATH  MathSciNet  Google Scholar 

  77. R. Roth,El origen de la teoría de grupos. El teorema de Lagrange (1771), Bol. Math.,3 (1969), 137–141.

    Google Scholar 

  78. D. Sachs,Identities in finite partition lattices, Proc. Amer. Math. Soc.12 (1969), 944–945.

    Article  MathSciNet  Google Scholar 

  79. B. M. Schein,A remark on bisimple inverse semigroups, Semigroup Forum3 (1971), 80–83.

    Article  MATH  MathSciNet  Google Scholar 

  80. A. Schmidt,Über deduktive Theorien mit mehreren Sorten von Grunddingen, Math. Ann.115 (1938), 485–506

    Article  MathSciNet  Google Scholar 

  81. A. Schmidt,Die Zulässigkeit der Behandlung mehrsortiger Theorien mittels der üblichen einsortigen Prädikatenlogik, Math. Ann.123 (1951), 187–200.

    Article  MATH  MathSciNet  Google Scholar 

  82. E. T. Schmidt,Kongruenzrelationen algebraischer Strukturen, Math. Forschungsberichte25, Berlin, 1969.

  83. E. T. Schmidt,Über reguläre Mannigfaltigkeiten, Acta Sci. Math. (Szeged)31 (1970), (197–201.

    MATH  MathSciNet  Google Scholar 

  84. E. T. Schmidt,On n-permutable equational classes, Acta Sci. Math. (Szeged) 33 (1972), 29–30.

    MATH  MathSciNet  Google Scholar 

  85. J. Słomiński,On the determining of the form of congruences in abstract algebras with equationally definable constant elements, Fund. Math.48 (1960), 325–341.

    MATH  MathSciNet  Google Scholar 

  86. S. K. Stein,Finite models of identities, Proc. Amer. Math. Soc.14 (1963), 216–222.

    Article  MATH  MathSciNet  Google Scholar 

  87. M. G. Stone,Proper congruences do not imply a modular congruence lattice, Colloq. Math.23 (1971), 25–27.

    MATH  MathSciNet  Google Scholar 

  88. S. Świerczkowski,On ismorphic free algebras, Fund. Math. 50 (1961) 35–44.

    MATH  MathSciNet  Google Scholar 

  89. A. Tarski,Equational logic and equational theories of algebras, p. 275–288, in: H.A. Schmidt, K. Schütte, and H.-J. Thiele (eds.), Contributions to mathematical logic, North-Holland, Amsterdam, 1968.

    Google Scholar 

  90. W. Taylor,Some constructions of compact algebras, Ann. Math. Logic3 (1971), 395–435.

    Article  MATH  MathSciNet  Google Scholar 

  91. W. Taylor,Fixed points of endomorphisms, Algebra Universalis2 (1972), 74–76.

    MATH  MathSciNet  Google Scholar 

  92. W. Taylor,Characterizing Mal’cev conditions, Abstract 72T-A98, Notices Amer. Math. Soc.19 (1972), A431.

    Google Scholar 

  93. W. Taylor,Varieties without doubleton algebras, Abstract 72T-A280, Notices Amer. Math. Soc.19 (1972), A-753.

    Google Scholar 

  94. H. A. Thurston,Derived operations and congruences, Proc. London Math. Soc. (3)8 (1958), 127–134.

    Article  MATH  MathSciNet  Google Scholar 

  95. B. A. TrakhtenbrotThe impossibility of an algorithm for the problem of satisfiability in finite classes, Dokl. Akad. Nauk S.S.S.R. (N.S.)70 (1950), 569–572.

    Google Scholar 

  96. I. I. Valutse,Universal algebras with regular but not permutable congruences (in Russian), Uspekh. Mat. Nauk18 (3) (1963), 145–148.

    Google Scholar 

  97. L. I. Wade,Post algebras and rings, Duke Math. J.12 (1945), 389–395.

    Article  MathSciNet  Google Scholar 

  98. H. Wang,Logic of many-sorted theories, J. Symbolic Logic,17 (1952), 105–116.

    Article  MATH  MathSciNet  Google Scholar 

  99. R. Wille,Kongruenzklassengeometrien, Lecture notes in mathematics #113, Springer-Verlag, Berlin, 1970.

    MATH  Google Scholar 

  100. G. C. Wraith,Algebraic theories, Aarhus University Lecture Notes Series No. 22, April, 1970.

  101. G. C. Wraith,Algebras over theories, Colloq. Math.23 (1971), 181–190.

    MATH  MathSciNet  Google Scholar 

Additional references

  1. J. Adámek and J. Reiterman,Fixed-point property of unary algebras, Algebra Universalis (to appear).

  2. O. Chein,Lagrange’s Theorem for M k-loops, Arch. Math. (Basel)24 (1973), 121–122.

    MATH  MathSciNet  Google Scholar 

  3. A. Day,p-Modularity implies modularity in equational classes, Algebra Universalis (to appear).

  4. E. Gedeonová,Acta F.R.N. Univ., Commen. Math.28 (1972), 99–106.

    MATH  Google Scholar 

  5. L. Klukovits,Hamiltonian varieties of universal algebras, Acta Sci. Math. (Szeged) (to appear).

  6. R. McKenzie,Simple undecidable problems about finite groupoids (preliminary report), Abstract 73T-E95, Notices Amer. Math. Soc.20 (1973), A-505.

    Google Scholar 

  7. A. Shafaat,Lattices of subsemivarieties of certain varieties, J. Austral. Math. Soc.,12 (1971), 15–20.

    Article  MATH  MathSciNet  Google Scholar 

  8. A. Ursini,Sulle varietà di algebre con una buona teoria degli ideali, Boll. Unione Mat. Ital. (4)6 (1972), 90–95.

    MATH  MathSciNet  Google Scholar 

  9. A. Ursini,Osservazioni sulle varietà BIT, Boll. Unione Mat. Ita. (4)7 (1973), 205–211.

    MATH  MathSciNet  Google Scholar 

  10. S. Bulman-Fleming, A. Day and W. Taylor,Regularity and modularity of congruences, Algebra Universalis (to appear).

  11. R. A. Dean,Component subsets of the free lattice on n generators, Proc. Amer. Math. Soc.7 (1956), 220–226.

    Article  MATH  MathSciNet  Google Scholar 

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Taylor, W. Characterizing Mal’cev conditions. Algebra Univ. 3, 351–397 (1973). https://doi.org/10.1007/BF02945141

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