Book Review
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MathSciNet review:
1567948
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Book Information:
Authors:
Christian U. Jensen and
Helmut Lenzing
Title:
Model theoretic algebra
Additional book information:
Gordon & Breach, New York, 1989, 430 pp., $59.00. ISBN 2-88124-717-2.
J. W. Addison, Leon Henkin, and Alfred Tarski (eds.), The theory of models, North-Holland Publishing Co., Amsterdam, 1965. MR 0195680
Anne Bauval, Polynomial rings and weak second-order logic, J. Symbolic Logic 50 (1985), no. 4, 953–972 (1986). MR 820125, DOI 10.2307/2273983
Greg Cherlin, Model theoretic algebra—selected topics, Lecture Notes in Mathematics, Vol. 521, Springer-Verlag, Berlin-New York, 1976. MR 0539999
Gert H. Müller, Wolfgang Lenski, and Wolfgang Rautenberg (eds.), $\Omega$-bibliography of mathematical logic. Vol. I, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1987. Classical logic. MR 915631
Michael D. Fried and Moshe Jarden, Field arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 11, Springer-Verlag, Berlin, 1986. MR 868860, DOI 10.1007/978-3-662-07216-5
Leon Henkin, Some interconnections between modern algebra and mathematical logic, Trans. Amer. Math. Soc. 74 (1953), 410–427. MR 55287, DOI 10.1090/S0002-9947-1953-0055287-X
Grete Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), no. 1, 736–788 (German). MR 1512302, DOI 10.1007/BF01206635
Edwin Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 26239, DOI 10.1090/S0002-9947-1948-0026239-9
Graham Higman, B. H. Neumann, and Hanna Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (1949), 247–254. MR 32641, DOI 10.1112/jlms/s1-24.4.247
10. A. I. Maltsev, On the faithful representation of infinite groups by matrices, Mat. Sb. (NS) 8 (50) (1940), 405-422. (Russian)
A. Mal′cev, On a general method for obtaining local theorems in group theory, Ivanov. Gos. Ped. Inst. Uč. Zap. Fiz.-Mat. Fak. 1 (1941), no. 1, 3–9 (Russian). MR 0075939
Anatoliĭ Ivanovič Mal′cev, The metamathematics of algebraic systems. Collected papers: 1936–1967, Studies in Logic and the Foundations of Mathematics, Vol. 66, North-Holland Publishing Co., Amsterdam-London, 1971. Translated, edited, and provided with supplementary notes by Benjamin Franklin Wells, III. MR 0349383
B. H. Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 4–11. MR 8808, DOI 10.1112/jlms/s1-18.1.4
Mike Prest, Model theory and modules, London Mathematical Society Lecture Note Series, vol. 130, Cambridge University Press, Cambridge, 1988. MR 933092, DOI 10.1017/CBO9780511600562
Abraham Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland Publishing Co., Amsterdam, 1963. MR 0153570
Abraham Robinson, Selected papers of Abraham Robinson. Vol. I, Yale University Press, New Haven, Conn., 1979. Model theory and algebra; Edited and with an introduction by H. Jerome Keisler; With a biography by George B. Seligman. MR 533887
Julia Robinson, The decision problem for fields, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 299–311. MR 0200163
W. Szmielew, Decision problem in group theory, Library of the Tenth International Congress of Philosophy, Amsterdam, August 11–18, 1948, Vol. I, Proceedings of the Congress, publisher unknown, 1949, pp. 763–766. MR 0028797
Gaisi Takeuti, Two applications of logic to mathematics, Kanô Memorial Lectures, Vol. 3, Iwanami Shoten Publishers, Tokyo; Princeton University Press, Princeton, N.J., 1978. MR 0505474
20. A. Tarski, Sur les ensembles définissables de nombres réels. I, Fund. Math. 17 (1931), 210-239.
Alfred Tarski, A Decision Method for Elementary Algebra and Geometry, The Rand Corporation, Santa Monica, Calif., 1948. MR 0028796
- 1.
- J. W. Addison, L. Henkin, and A. Tarski (eds. ), The theory of models, Proceedings of the 1963 International Symposium at Berkeley, North-Holland, Amsterdam, 1965. MR 0195680
- 2.
- A. Bauval, Polynomial rings and weak second-order logic, J. Symbolic Logic 50 (1985), 953-972. MR 820125
- 3.
- G. L. Cherlin, Model theoretic algebra, Selected Topics. Lecture Notes in Math., vol. 521, Springer-Verlag, Berlin, 1976. MR 539999
- 4.
- H. D. Ebbinghaus (ed. ), Ω-bibliography of mathematical logic, vol. III, Model Theory, Springer-Verlag, Berlin, 1987. MR 915633
- 5.
- M. Fried and M. Jarden, Field arithmetic, Springer-Verlag, Berlin, 1986. MR 868860
- 6.
- L. Henkin, Some interconnections between modern algebra and mathematical logic, Trans. Amer. Math. Soc. 74 (1953), 410-427. MR 55287
- 7.
- G. Hermann, Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann. 95 (1926), 736-788. MR 1512302
- 8.
- E. Hewitt, Rings of real-valued continuous functions, Trans. Amer. Math. Soc. 64 (1948), 45-99. MR 26239
- 9.
- G. Higman, B. H. Neumann, and H. Neumann, Embedding theorems for groups, J. London Math. Soc. 24 (149), 247-254. MR 32641
- 10.
- A. I. Maltsev, On the faithful representation of infinite groups by matrices, Mat. Sb. (NS) 8 (50) (1940), 405-422. (Russian)
- 11.
- A. I. Maltsev, A general method for obtaining local theorems in group theory, Uch Zap Ped Inst., Ivanovo 1 (1941), 3-9. (Russian) MR 75939
- 12.
- A. I. Maltsev, The metamathematics of algebraic systems, Collected papers: 1936-1967, North-Holland, Amsterdam, 1971. MR 349383
- 13.
- B. H. Neumann, Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 4-11. MR 8808
- 14.
- M. Prest, Model theory and modules, London Mathematical Society Lecture Notes Series 130, Cambridge Univ. Press, Cambridge, 1988. MR 933092
- 15.
- A. Robinson, Introduction to model theory and to the metamathematics of algebra, North-Holland, Amsterdam, 1963. MR 153570
- 16.
- A. Robinson, Selected papers, vol. I: Model Theory and Algebra, Yale Univ. Press, New Haven, 1979. MR 533887
- 17.
- J. Robinson, The decision problem for fields, Proceedings of the 1963 International Symposium at Berkeley, pp. 299-311, North-Holland, Amsterdam, 1965. MR 200163
- 18.
- W. Szmielew, Decision problem in group theory, Internat. Congr. Philos., 1948, Amsterdam, North-Holland, Amsterdam, 1949, pp. 763-766. MR 28797
- 19.
- G. Takeuti, Two applications of logic to mathematics, Princeton Univ. Press, Princeton, NJ, 1978. MR 505474
- 20.
- A. Tarski, Sur les ensembles définissables de nombres réels. I, Fund. Math. 17 (1931), 210-239.
- 21.
- A. Tarski, A decision method for elementary algebra and geometry, Rand Corporation, Santa Monica, 1948. MR 28796
Review Information:
Reviewer:
G. Sabbagh
Journal:
Bull. Amer. Math. Soc.
25 (1991), 215-221
DOI:
https://doi.org/10.1090/S0273-0979-1991-16070-2
