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Primitive satisfaction and equational problems for lattices and other algebras


Author: Kirby A. Baker
Journal: Trans. Amer. Math. Soc. 190 (1974), 125-150
MSC: Primary 08A15; Secondary 06A70
DOI: https://doi.org/10.1090/S0002-9947-1974-0349532-4
MathSciNet review: 0349532
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Abstract | References | Similar Articles | Additional Information

Abstract: This paper presents a general method of solving equational problems in all equational classes of algebras whose congruence lattices are distributive, such as those consisting of lattices, relation algebras, cylindric algebras, orthomodular lattices, lattice-ordered rings, lattice-ordered groups, Heyting algebras, other lattice-ordered algebras, implication algebras, arithmetic rings, and arithmetical algebras.


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  • [1] K. A. Baker, Equational classes of modular lattices, Pacific J. Math. 28 (1969), 9-15. MR 39 #5435. MR 0244118 (39:5435)
  • [2] -, Equational axioms for classes of lattices, Bull. Amer. Math. Soc. 77 (1971), 97-102. MR 44 #5256. MR 0288058 (44:5256)
  • [3] -, Congruence-valued logic (to appear).
  • [4] -, Equational axioms for classes of Heyting algebras (preprint),
  • [5] G. Birkhoff, On the structure of abstract algebras, Proc. Cambridge Philos. Soc. 31 (1935), 433-454.
  • [6] -, Subdirect unions in universal algebra, Bull. Amer. Math. Soc. 50 (1944), 764-768. MR 6, 33. MR 0010542 (6:33d)
  • [7] -, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc., Providence, R. I., 1967. MR 37 #2638. MR 0227053 (37:2638)
  • [8] G. Bruns and G. Kalmbach, Varieties of orthomodular lattices, Canad. J. Math. 23 (1971), 802-810. MR 44 #6565. MR 0289374 (44:6565)
  • [9] -, Varieties of orthomodular lattices. II, Canad. J. Math. 24 (1972), 328-337. MR 45 #3267. MR 0294194 (45:3267)
  • [10] G. Epstein, The lattice theory of Post algebras, Trans. Amer. Math. Soc. 95 (1960), 300-317. MR 22 #3701. MR 0112855 (22:3701)
  • [11] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York; Addison-Wesley, Reading, Mass., 1963. MR 30 #2090. MR 0171864 (30:2090)
  • [12] G. Grätzer, Universal algebra, Van Nostrand, Princeton, N. J., 1968. MR 40 #1320. MR 0248066 (40:1320)
  • [13] -, Lattice theory: First concepts and distributive lattices, Freeman, San Francisco, 1971. MR 0321817 (48:184)
  • [14] -, Personal communication.
  • [15] L. Henkin and A. Tarski, Cylindric algebras, Proc. Sympos. Pure Math., vol. 2, Amer. Math. Soc., Providence, R. I., 1961, pp. 83-113. MR 23 #A1564. MR 0124250 (23:A1564)
  • [16] L. Henkin, D. Monk and A. Tarski, Cylindric algebras. Part I, North-Holland, Amsterdam, 1971.
  • [17] C. Herrmann, Weak (projective) radius and finite equational bases for classes of lattices, Preprint no. 29, Aug. 1972, Technische Hochschule Darmstadt. MR 0329983 (48:8322)
  • [18] S. S. Holland, Jr., Current interest in orthomodular lattices, Trends in lattice theory (Sympos., U. S. Naval Academy, Annapolis, Md., 1966), J. C. Abbott, Ed., Van Nostrand-Reinhold, New York, 1970, pp. 41-126. MR 42 #7569. MR 0272688 (42:7569)
  • [19] J. R. Isbell, Notes on ordered rings, Algebra Universalis 1 (1971/72), 393-399. MR 45 #5055. MR 0295994 (45:5055)
  • [20] B. Jónsson, Algebras whose congruence lattices are distributive, Math. Scand. 21 (1967), 110-121. MR 38 #5689. MR 0237402 (38:5689)
  • [21] -, Topics in universal algebra, Lecture Notes, Vanderbilt University, Nashville, Tenn., (1969/70).
  • [22] B. Jónsson and A. Tarski, Boolean algebras with operators. II, Amer. J. Math. 74 (1952), 127-162. MR 13, 524. MR 0045086 (13:524g)
  • [23] R. C. Lyndon, Identities in finite algebras, Proc. Amer. Math. Soc. 5 (1954), 8-9. MR 15, 676. MR 0060482 (15:676b)
  • [24] A. I. Mal'cev, On the general theory of algebraic systems, Mat. Sb. 35 (77) (1954), 3-20; English transl., Amer. Math. Soc. Transl. (2) 27 (1963), 125-142. MR 27 #1401. MR 0151416 (27:1401)
  • [25] C. G. McKay, On finite logics, Nederl. Akad. Wetensch. Proc. Ser. A 70 = Indag. Math. 29 (1967), 363-365. MR 35 #6524. MR 0215689 (35:6524)
  • [26] R. McKenzie, Equational bases for lattice theories, Math. Scand. 27 (1970), 24-38. MR 43 #118. MR 0274353 (43:118)
  • [27] G. Michler and R. Wille, Die primitiven Klassen arithmetischer Ringe, Math. Z. 113 (1970), 369-372. MR 41 #5420. MR 0260797 (41:5420)
  • [28] A. Mitschke, Implication algebras are 3-permutable and 3-distributive, Algebra Universalis 1 (1971/72), 182-186. MR 0309828 (46:8933)
  • [29] D. Monk, On equational classes of algebraic versions of logic. I, Math. Scand. 27 (1970), 53-71. MR 43 #6065. MR 0280345 (43:6065)
  • [30] -, Personal communication.
  • [31] S. Oates MacDonald, Various varieties, Pure Mathematics Preprint, no. 26, Dept. of Math., University of Queensland, Brisbane, Australia, 1972. MR 0354494 (50:6972)
  • [32] O. Ore, On the theorem of Jordan-Hölder, Trans. Amer. Math. Soc. 41 (1937), 266-275. MR 1501901
  • [33] P. Perkins, Bases for equational theories of semigroups, J. Algebra 11 (1969), 293-314. MR 38 #2232. MR 0233911 (38:2232)
  • [34] R. S. Pierce, Modules over commutative regular rings, Mem. Amer. Math. Soc. No. 70 (1967). MR 36 #151. MR 0217056 (36:151)
  • [35] A. F. Pixley, Distributivity and permutability of congruence relations in equational classes of algebras, Proc. Amer. Math. Soc. 14 (1963), 105-109. MR 26 #3630. MR 0146104 (26:3630)
  • [36] -, Completeness in arithmetical algebras, Algebra Universalis 2 (1972), 179-196. MR 0321843 (48:208)
  • [37] R. Quackenbush, personal communication.
  • [38] H. Rasiowa and R. Sikorski, The mathematics of metamathematics, 3rd ed., Monografia Matematyczne t. 41, Warszawa, 1970. MR 0344067 (49:8807)
  • [39] E. T. Schmidt, Kongruenzrelationen algebraischer Strukturen, Math. Forschungsber. 25, VEB Deutscher Verlag der Wissenschaften, Berlin, 1969. MR 41 #136. MR 0255474 (41:136)
  • [40] A. Tarski, On the calculus of relations, J. Symbolic Logic 6 (1941), 73-89. MR 3, 130. MR 0005280 (3:130e)
  • [41] -, Equational logic and equational theories of algebras, Contributions to Math. Logic (Colloq., Hannover,. 1966), H. A. Schmidt et al., Eds., North-Holland, Amsterdam, 1968, pp. 275-288. MR 38 #5692. MR 0237410 (38:5692)
  • [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. MR 28 #5015. MR 0161811 (28:5015)
  • [43] H. Werner and R. Wille, Charakterisierung der primitiven Klassen arithmetischer Ringe, Math. Z. 115 (1970), 197-200. MR 42 #318. MR 0265408 (42:318)
  • [44] R. Wille, Primitive Länge und primitive Weite bei modularen Verbänden, Math. Z. 108 (1969), 129-136. MR 39 #2672. MR 0241332 (39:2672)
  • [45] -, Variety invariants for modular lattices, Canad. J. Math. 21 (1969), 279-283. MR 39 #2671. MR 0241331 (39:2671)
  • [46] -, Kongruenzklassengeometrien, Lecture Notes in Math. vol. 113, Springer-Verlag, Berlin and New York, 1970. MR 41 #6759. MR 0262149 (41:6759)
  • [47] -, Primitive subsets of lattices, Algebra Universalis 2 (1972), 95-98. MR 0311524 (47:86)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0349532-4
Keywords: Equational problem, equational class, variety, congruence lattice, lattice, lattice-ordered algebra
Article copyright: © Copyright 1974 American Mathematical Society

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