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Primitive satisfaction and equational problems for lattices and other algebras. (English) Zbl 0291.08001


MSC:

08B99 Varieties
06F25 Ordered rings, algebras, modules
06B05 Structure theory of lattices
06D05 Structure and representation theory of distributive lattices
06E05 Structure theory of Boolean algebras
06F15 Ordered groups
08C10 Axiomatic model classes
08Axx Algebraic structures
03B99 General logic
03G15 Cylindric and polyadic algebras; relation algebras
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[1] Kirby A. Baker, Equational classes of modular lattices, Pacific J. Math. 28 (1969), 9 – 15. · Zbl 0174.29802
[2] Kirby A. Baker, Equational axioms for classes of lattices, Bull. Amer. Math. Soc. 77 (1971), 97 – 102. · Zbl 0209.31901
[3] -, Congruence-valued logic (to appear).
[4] -, Equational axioms for classes of Heyting algebras (preprint) · Zbl 0355.02039
[5] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454. · Zbl 0013.00105
[6] Garrett Birkhoff, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764 – 768. · Zbl 0060.05809
[7] Garrett Birkhoff, Lattice theory, Third edition. American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. · Zbl 0153.02501
[8] Günter Bruns and Gudrun Kalmbach, Varieties of orthomodular lattices, Canad. J. Math. 23 (1971), 802 – 810. · Zbl 0278.06003
[9] Günter Bruns and Gudrun Kalmbach, Varieties of orthomodular lattices. II, Canad. J. Math. 24 (1972), 328 – 337. · Zbl 0278.06004
[10] George Epstein, The lattice theory of Post algebras, Trans. Amer. Math. Soc. 95 (1960), 300 – 317. · Zbl 0207.29403
[11] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. · Zbl 0137.02001
[12] George Grätzer, Universal algebra, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1968. · Zbl 0155.03502
[13] George Grätzer, Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. · Zbl 0232.06001
[14] -, Personal communication.
[15] Leon Henkin and Alfred Tarski, Cylindric algebras, Proc. Sympos. Pure Math., Vol. II, American Mathematical Society, Providence, R.I., 1961, pp. 83 – 113. · Zbl 0576.03043
[16] L. Henkin, D. Monk and A. Tarski, Cylindric algebras. Part I, North-Holland, Amsterdam, 1971. · Zbl 0214.01302
[17] C. Herrmann, Weak (projective) radius and finite equational bases for classes of lattices, Algebra Universalis 3 (1973), 51 – 58. · Zbl 0288.06008
[18] Samuel S. Holland Jr., The current interest in orthomodular lattices, Trends in Lattice Theory (Sympos., U.S. Naval Academy, Annapolis, Md., 1966), Van Nostrand Reinhold, New York, 1970, pp. 41 – 126.
[19] J. R. Isbell, Notes on ordered rings, Algebra Universalis 1 (1971/72), 393 – 399. · Zbl 0238.06013
[20] Bjarni Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110 – 121 (1968). · Zbl 0167.28401
[21] -, Topics in universal algebra, Lecture Notes, Vanderbilt University, Nashville, Tenn., (1969/70).
[22] Bjarni Jónsson and Alfred Tarski, Boolean algebras with operators. II, Amer. J. Math. 74 (1952), 127 – 162. · Zbl 0045.31601
[23] R. C. Lyndon, Identities in finite algebras, Proc. Amer. Math. Soc. 5 (1954), 8 – 9. · Zbl 0055.02705
[24] A. I. Mal\(^{\prime}\)cev, On the general theory of algebraic systems, Amer. Math. Soc. Transl. (2) 27 (1963), 125 – 142.
[25] C. G. McKay, On finite logics, Nederl. Akad. Wetensch. Proc. Ser. A 70=Indag. Math. 29 (1967), 363 – 365. · Zbl 0153.00702
[26] Ralph McKenzie, Equational bases for lattice theories, Math. Scand. 27 (1970), 24 – 38. · Zbl 0307.08001
[27] Gerhard Michler and Rudolf Wille, Die primitiven Klassen arithmetischer Ringe, Math. Z. 113 (1970), 369 – 372 (German). · Zbl 0207.04703
[28] Aleit Mitschke, Implication algebras are 3-permutable and 3-distributive, Algebra Universalis 1 (1971/72), 182 – 186. · Zbl 0242.08005
[29] Donald Monk, On equational classes of algebraic versions of logic. I, Math. Scand. 27 (1970), 53 – 71. · Zbl 0208.01202
[30] -, Personal communication.
[31] Sheila Oates MacDonald, Various varieties, J. Austral. Math. Soc. 16 (1973), 363 – 367. Collection of articles dedicated to the memory of Hanna Neumann, III. · Zbl 0272.08005
[32] Oystein Ore, On the theorem of Jordan-Hölder, Trans. Amer. Math. Soc. 41 (1937), no. 2, 266 – 275. · JFM 63.0083.02
[33] Peter Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 298 – 314. · Zbl 0186.03401
[34] R. S. Pierce, Modules over commutative regular rings, Memoirs of the American Mathematical Society, No. 70, American Mathematical Society, Providence, R.I., 1967. · Zbl 0152.02601
[35] A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963), 105 – 109. · Zbl 0113.24804
[36] A. F. Pixley, Completeness in arithmetical algebras, Algebra Universalis 2 (1972), 179 – 196. · Zbl 0254.08010
[37] R. Quackenbush, personal communication.
[38] Helena Rasiowa and Roman Sikorski, The mathematics of metamathematics, 3rd ed., PWN — Polish Scientific Publishers, Warsaw, 1970. Monografie Matematyczne, Tom 41. · Zbl 0239.02002
[39] E. T. Schmidt, Kongruenzrelationen algebraischer Strukturen, Mathematische Forschungsberichte, XXV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1969 (German). · Zbl 0198.33301
[40] Alfred Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 73 – 89. · Zbl 0026.24401
[41] A. Tarski, Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 275 – 288.
[42] T. Traczyk, An equational definition of a class of Post algebras, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 12 (1964), 147 – 149. · Zbl 0171.25702
[43] Heinrich Werner and Rudolf Wille, Charakterisierungen der primitiven Klassen arithmetischer Ringe, Math. Z. 115 (1970), 197 – 200 (German). · Zbl 0216.33701
[44] Rudolf Wille, Primitive Länge und primitive Weite bei modularen Verbänden, Math. Z. 108 (1969), 129 – 136 (German). · Zbl 0169.32403
[45] Rudolf Wille, Variety invariants for modular lattices, Canad. J. Math. 21 (1969), 279 – 283. · Zbl 0208.29102
[46] Rudolf Wille, Kongruenzklassengeometrien, Lecture Notes in Mathematics, Vol. 113, Springer-Verlag, Berlin-New York, 1970 (German). · Zbl 0191.51403
[47] R. Wille, Primitive subsets of lattices, Algebra Universalis 2 (1972), 95 – 98. · Zbl 0269.06001
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