Existentially complete abelian lattice-ordered groups
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- by A. M. W. Glass and Keith R. Pierce PDF
- Trans. Amer. Math. Soc. 261 (1980), 255-270 Request permission
Abstract:
The theory of abelian totally ordered groups has a model completion. We show that the theory of abelian lattice-ordered groups has no model companion. Indeed, the Archimedean property can be captured by a first order $\forall \exists \forall$ sentence for existentially complete abelian lattice-ordered groups, and distinguishes between finitely generic abelian lattice-ordered groups and infinitely generic ones. We then construct (by sheaf techniques) the model companions of certain classes of discrete abelian lattice-ordered groups.References
- Alain Bigard, Klaus Keimel, and Samuel Wolfenstein, Groupes et anneaux réticulés, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR 0552653
- Greg Cherlin, Model theoretic algebra—selected topics, Lecture Notes in Mathematics, Vol. 521, Springer-Verlag, Berlin-New York, 1976. MR 0539999
- Stephen D. Comer, Complete and model-complete theories of monadic algebras, Colloq. Math. 34 (1975/76), no. 2, 183–190. MR 480008, DOI 10.4064/cm-34-2-183-190
- Paul F. Conrad, Regularly ordered groups, Proc. Amer. Math. Soc. 13 (1962), 726–731. MR 146272, DOI 10.1090/S0002-9939-1962-0146272-9 —, Lattice-ordered groups, Tulane Lecture Notes, Tulane Univ., New Orleans, 1970.
- John Dauns and Karl Heinrich Hofmann, The representation of biregular rings by sheaves, Math. Z. 91 (1966), 103–123. MR 186693, DOI 10.1007/BF01110158
- Paul Eklof and Gabriel Sabbagh, Model-completions and modules, Ann. Math. Logic 2 (1970/71), no. 3, 251–295. MR 277372, DOI 10.1016/0003-4843(71)90016-7 George Eliot, Middlemarch, 1872.
- A. M. W. Glass and Keith R. Pierce, Equations and inequations in lattice-ordered groups, Ordered groups (Proc. Conf., Boise State Univ., Boise, Idaho, 1978), Lecture Notes in Pure and Appl. Math., vol. 62, Dekker, New York, 1980, pp. 141–171. MR 601623
- Joram Hirschfeld and William H. Wheeler, Forcing, arithmetic, division rings, Lecture Notes in Mathematics, Vol. 454, Springer-Verlag, Berlin-New York, 1975. MR 0389581
- L. Lipshitz and D. Saracino, The model companion of the theory of commutative rings without nilpotent elements, Proc. Amer. Math. Soc. 38 (1973), 381–387. MR 439624, DOI 10.1090/S0002-9939-1973-0439624-8
- Angus Macintyre, Model-completeness for sheaves of structures, Fund. Math. 81 (1973/74), no. 1, 73–89. MR 337592, DOI 10.4064/fm-81-1-73-89
- Model theory, Handbook of mathematical logic, Part A, Studies in Logic and the Foundations of Math., Vol. 90, North-Holland, Amsterdam, 1977, pp. 3–313. With contributions by Jon Barwise, H. Jerome Keisler, Paul C. Eklof, Angus Macintyre, Michael Morley, K. D. Stroyan, M. Makkai, A. Kock and G. E. Reyes. MR 0491125
- Keith R. Pierce, Amalgamations of lattice ordered groups, Trans. Amer. Math. Soc. 172 (1972), 249–260. MR 325488, DOI 10.1090/S0002-9947-1972-0325488-3
- Abraham Robinson, Complete theories, North-Holland Publishing Co., Amsterdam, 1956. MR 0075897
- Abraham Robinson and Elias Zakon, Elementary properties of ordered abelian groups, Trans. Amer. Math. Soc. 96 (1960), 222–236. MR 114855, DOI 10.1090/S0002-9947-1960-0114855-0
- D. Saracino, Existentially complete nilpotent groups, Israel J. Math. 25 (1976), no. 3-4, 241–248. MR 453518, DOI 10.1007/BF02757003
- D. Saracino, Existentially complete torsion-free nilpotent groups, J. Symbolic Logic 43 (1978), no. 1, 126–134. MR 480011, DOI 10.2307/2271955
- Dan Saracino and Carol Wood, Periodic existentially closed nilpotent groups, J. Algebra 58 (1979), no. 1, 189–207. MR 535853, DOI 10.1016/0021-8693(79)90199-6
- William H. Wheeler, A characterization of companionable, universal theories, J. Symbolic Logic 43 (1978), no. 3, 402–429. MR 491131, DOI 10.2307/2273518
Additional Information
- © Copyright 1980 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 261 (1980), 255-270
- MSC: Primary 03C60; Secondary 03C35, 06F20
- DOI: https://doi.org/10.1090/S0002-9947-1980-0576874-2
- MathSciNet review: 576874