Existentially complete abelian lattice-ordered groups

Glass, A. M. W.; Pierce, Keith R.

doi:10.1090/S0002-9947-1980-0576874-2

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

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

Middlemarch

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