Spectral spaces of countable Abelian lattice-ordered groups
HTML articles powered by AMS MathViewer
- by Friedrich Wehrung PDF
- Trans. Amer. Math. Soc. 371 (2019), 2133-2158 Request permission
Abstract:
It is well known that the $\ell$-spectrum of an Abelian $\ell$-group, defined as the set of all its prime $\ell$-ideals with the hull-kernel topology, is a completely normal generalized spectral space. We establish the following converse of this result.
Theorem. Every second countable, completely normal generalized spectral space is homeomorphic to the $\ell$-spectrum of some Abelian $\ell$-group.
We obtain this result by proving that a countable distributive lattice $D$ with zero is isomorphic to the Stone dual of some $\ell$-spectrum (we say that $D$ is $\ell$-representable) iff for all $a,b\in D$ there are $x,y\in D$ such that $a\vee b=a\vee y=b\vee x$ and $x\wedge y=0$. On the other hand, we construct a non-$\ell$-representable bounded distributive lattice, of cardinality $\aleph _1$, with an $\ell$-representable countable $\mathscr {L}_{\infty ,\omega }$-elementary sublattice. In particular, there is no characterization, of the class of all $\ell$-representable distributive lattices, by any class of $\mathscr {L}_{\infty ,\omega }$ sentences.
References
- Kirby A. Baker, Free vector lattices, Canadian J. Math. 20 (1968), 58–66. MR 224524, DOI 10.4153/CJM-1968-008-x
- Jon Barwise, Admissible sets and structures, Perspectives in Mathematical Logic, Springer-Verlag, Berlin-New York, 1975. An approach to definability theory. MR 0424560
- John L. Bell, Infinitary logic, The Stanford Encyclopedia of Philosophy (Edward N. Zalta, ed.), Metaphysics Research Lab, Stanford University, winter 2016 ed., 2016, accessible at the URL https://plato.stanford.edu/entries/logic-infinitary/.
- George M. Bergman, Von Neumann regular rings with tailor-made ideal lattices, Unpublished note, available online at http://math.berkeley.edu/\~{}gbergman/papers/unpub/, October 26, 1986.
- 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
- R. Cignoli, D. Gluschankof, and F. Lucas, Prime spectra of lattice-ordered abelian groups, J. Pure Appl. Algebra 136 (1999), no. 3, 217–229. MR 1675803, DOI 10.1016/S0022-4049(98)00031-0
- Roberto Cignoli and Antoni Torrens, The poset of prime $l$-ideals of an abelian $l$-group with a strong unit, J. Algebra 184 (1996), no. 2, 604–612. MR 1409232, DOI 10.1006/jabr.1996.0278
- Michel Coste and Marie-Françoise Roy, La topologie du spectre réel, Ordered fields and real algebraic geometry (San Francisco, Calif., 1981), Contemp. Math., vol. 8, Amer. Math. Soc., Providence, R.I., 1982, pp. 27–59 (French). MR 653174
- B. A. Davey and H. A. Priestley, Introduction to lattices and order, Cambridge Mathematical Textbooks, Cambridge University Press, Cambridge, 1990. MR 1058437
- Charles N. Delzell and James J. Madden, A completely normal spectral space that is not a real spectrum, J. Algebra 169 (1994), no. 1, 71–77. MR 1296582, DOI 10.1006/jabr.1994.1272
- Charles N. Delzell and James J. Madden, Lattice-ordered rings and semialgebraic geometry. I, Real analytic and algebraic geometry (Trento, 1992) de Gruyter, Berlin, 1995, pp. 103–129. MR 1320313
- Antonio Di Nola and Revaz Grigolia, Pro-finite MV-spaces, Discrete Math. 283 (2004), no. 1-3, 61–69. MR 2061482, DOI 10.1016/j.disc.2003.12.015
- M. A. Dickmann, Applications of model theory to real algebraic geometry. A survey, Methods in mathematical logic (Caracas, 1983) Lecture Notes in Math., vol. 1130, Springer, Berlin, 1985, pp. 76–150. MR 799038, DOI 10.1007/BFb0075308
- George A. Elliott and Daniele Mundici, A characterisation of lattice-ordered abelian groups, Math. Z. 213 (1993), no. 2, 179–185. MR 1221712, DOI 10.1007/BF03025717
- K. R. Goodearl, Partially ordered abelian groups with interpolation, Mathematical Surveys and Monographs, vol. 20, American Mathematical Society, Providence, RI, 1986. MR 845783, DOI 10.1090/surv/020
- K. R. Goodearl and F. Wehrung, Representations of distributive semilattices in ideal lattices of various algebraic structures, Algebra Universalis 45 (2001), no. 1, 71–102. MR 1809858, DOI 10.1007/s000120050203
- George Grätzer, Lattice theory. First concepts and distributive lattices, W. H. Freeman and Co., San Francisco, Calif., 1971. MR 0321817
- George Grätzer, Lattice theory: foundation, Birkhäuser/Springer Basel AG, Basel, 2011. MR 2768581, DOI 10.1007/978-3-0348-0018-1
- Wolf Iberkleid, Jorge Martínez, and Warren Wm. McGovern, Conrad frames, Topology Appl. 158 (2011), no. 14, 1875–1887. MR 2823701, DOI 10.1016/j.topol.2011.06.024
- Peter T. Johnstone, Stone spaces, Cambridge Studies in Advanced Mathematics, vol. 3, Cambridge University Press, Cambridge, 1982. MR 698074
- Carol R. Karp, Finite-quantifier equivalence, Theory of Models (Proc. 1963 Internat. Sympos. Berkeley), North-Holland, Amsterdam, 1965, pp. 407–412. MR 0209132
- Klaus Keimel, The representation of lattice-ordered groups and rings by sections in sheaves, Lectures on the applications of sheaves to ring theory (Tulane Univ. Ring and Operator Theory Year, 1970–1971, Vol. III), Lecture Notes in Math., Vol. 248, Springer, Berlin, 1971, pp. 1–98. MR 0422107
- Klaus Keimel, Some trends in lattice-ordered groups and rings, Lattice theory and its applications (Darmstadt, 1991) Res. Exp. Math., vol. 23, Heldermann, Lemgo, 1995, pp. 131–161. MR 1366870
- H. Jerome Keisler and Julia F. Knight, Barwise: infinitary logic and admissible sets, Bull. Symbolic Logic 10 (2004), no. 1, 4–36. MR 2062240, DOI 10.2178/bsl/1080330272
- David Kenoyer, Recognizability in the lattice of convex $l$-subgroups of a lattice-ordered group, Czechoslovak Math. J. 34(109) (1984), no. 3, 411–416. MR 761423
- Vincenzo Marra and Daniele Mundici, Combinatorial fans, lattice-ordered groups, and their neighbours: a short excursion, Sém. Lothar. Combin. 47 (2001/02), Art. B47f, 19. MR 1894026
- Vincenzo Marra and Daniele Mundici, MV-algebras and abelian $l$-groups: a fruitful interaction, Ordered algebraic structures, Dev. Math., vol. 7, Kluwer Acad. Publ., Dordrecht, 2002, pp. 57–88. MR 2083034, DOI 10.1007/978-1-4757-3627-4_{4}
- Stephen H. McCleary, Lattice-ordered groups whose lattices of convex $l$-subgroups guarantee noncommutativity, Order 3 (1986), no. 3, 307–315. MR 878927, DOI 10.1007/BF00400294
- Timothy Mellor and Marcus Tressl, Non-axiomatizability of real spectra in $\scr L_{\infty \lambda }$, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 2, 343–358 (English, with English and French summaries). MR 2978098
- António A. Monteiro, L’arithmétique des filtres et les espaces topologiques, Segundo symposium sobre algunos problemas matemáticos que se están estudiando en Latino América, Julio, 1954, Centro de Cooperación Científica de la UNESCO para América Latina, Montevideo, Uruguay, 1954, pp. 129–162 (French). MR 0074805
- D. Mundici, Advanced Łukasiewicz calculus and MV-algebras, Trends in Logic—Studia Logica Library, vol. 35, Springer, Dordrecht, 2011. MR 2815182, DOI 10.1007/978-94-007-0840-2
- Pavel Růžička, Jiří Tůma, and Friedrich Wehrung, Distributive congruence lattices of congruence-permutable algebras, J. Algebra 311 (2007), no. 1, 96–116. MR 2309879, DOI 10.1016/j.jalgebra.2006.11.005
- Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR 874114
- N. Schwartz, Real closed rings, Algebra and order (Luminy-Marseille, 1984) Res. Exp. Math., vol. 14, Heldermann, Berlin, 1986, pp. 175–194. MR 891460
- Marshall H. Stone, Topological representations of distributive lattices and Brouwerian logics, Čas. Mat. Fys. 67 (1938), no. 1, 1–25.
- F. Wehrung, The dimension monoid of a lattice, Algebra Universalis 40 (1998), no. 3, 247–411. MR 1668068, DOI 10.1007/s000120050091
- F. Wehrung, Semilattices of finitely generated ideals of exchange rings with finite stable rank, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1957–1970. MR 2031048, DOI 10.1090/S0002-9947-03-03369-5
- F. Wehrung, Real spectrum versus $\ell$-spectrum via Brumfiel spectrum, hal-01550450, preprint, July 2017.
Additional Information
- Friedrich Wehrung
- Affiliation: LMNO, CNRS UMR 6139, Département de Mathématiques, Université de Caen Normandie, 14032 Caen Cedex, France
- MR Author ID: 242737
- Email: friedrich.wehrung01@unicaen.fr
- Received by editor(s): April 6, 2017
- Received by editor(s) in revised form: September 26, 2017
- Published electronically: October 23, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 2133-2158
- MSC (2010): Primary 06D05, 06D20, 06D35, 06D50, 06F20, 46A55, 52A05, 52C35
- DOI: https://doi.org/10.1090/tran/7596
- MathSciNet review: 3894048