logo AFST

Non-axiomatizability of real spectra in λ
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 2, pp. 343-358.

Nous montrons que la propriété d’un espace spectral d’être un sous-espace spectral du spectre réel d’un anneau commutatif n’est pas exprimable dans le langage infinitaire du premier ordre λ de son treillis de définition. Ceci généralise un résultat de Delzell et Madden qui dit qu’en général, un espace spectral complètement normal n’est pas un spectre réel.

We show that the property of a spectral space, to be a spectral subspace of the real spectrum of a commutative ring, is not expressible in the infinitary first order language λ of its defining lattice. This generalises a result of Delzell and Madden which says that not every completely normal spectral space is a real spectrum.

Reçu le : 2011-01-07
Accepté le : 2011-11-30
Publié le : 2014-02-13
DOI : https://doi.org/10.5802/afst.1337
@article{AFST_2012_6_21_2_343_0,
     author = {Timothy Mellor and Marcus Tressl},
     title = {Non-axiomatizability of real spectra in $\mathcal{L}\_\infty \lambda $},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {343--358},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 21},
     number = {2},
     year = {2012},
     doi = {10.5802/afst.1337},
     zbl = {1254.03075},
     mrnumber = {2978098},
     language = {en},
     url = {afst.centre-mersenne.org/item/AFST_2012_6_21_2_343_0/}
}
Timothy Mellor; Marcus Tressl. Non-axiomatizability of real spectra in $\mathcal{L}_\infty \lambda $. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 21 (2012) no. 2, pp. 343-358. doi : 10.5802/afst.1337. https://afst.centre-mersenne.org/item/AFST_2012_6_21_2_343_0/

[1] Bochnak (J.), Coste (M.), Roy (M.-F.).— Real Algebraic Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 36, Springer (1998). | MR 1659509 | Zbl 0912.14023

[2] Carral (M.), Coste (M.).— Normal spectral spaces and their dimensions,J. Pure Appl. Algebra 30, no. 3, p. 227-235 (1983). | MR 724034 | Zbl 0525.14015

[3] Delzell (C.), Madden (J.).— A completely normal spectral space that is not a real spectrum, Journal of Algebra 169, p. 71-77 (1994). | MR 1296582 | Zbl 0833.14030

[4] Dickmann (M.).— Larger Infinitary Languages. Ch. IX of ’Model Theoretic Logics’ (J. Barwise, S. Feferman, eds.), Perspectives in Mathematical Logic, Springer-Verlag, Berlin (1985). | MR 819540 | Zbl 0324.02010

[5] Dickmann (M.), Gluschankof (D.), Lucas (F.).— The order structure of the real spectrum of commutative rings, J. Algebra 229, no. 1, p. 175-204 (2000). | MR 1765778

[6] Hochster (M.).— Prime ideal structure in commutative rings, Trans. Amer. Math. Soc. 142, p. 43-60 (1969). | MR 251026 | Zbl 0184.29401

[7] Hodges (W.).— Model Theory, Encyclopedia of mathematics and its applications, vol. 42 (1993). | MR 1221741 | Zbl 0789.03031

[8] Johnstone (P.T.).— Stone Spaces, Reprint of the 1982 edition. Cambridge Studies in Advanced Mathematics, 3. Cambridge University Press, Cambridge. xxii+370 pp. (1986) | MR 861951 | Zbl 0499.54001

[9] Kaplansky (I.).— Commutative RingsRevised edition. The University of Chicago Press, Chicago, Ill.-London, ix+182 pp. (1974) | MR 345945 | Zbl 0296.13001

[10] Priestley (H.A.).— Representation of distributive lattices by means of ordered stone spaces Bull. London Math. Soc. 2, p. 186-190 1970. | MR 265242 | Zbl 0201.01802

[11] Priestley (H.A.).— Spectral Sets, J. Pure Appl. Algebra 94, no. 1, p. 101-114 (1994). | MR 1277526 | Zbl 0807.06001

[12] Schwartz (N.).— The basic theory of real closed spacesMem. Am. Math. Soc., vol 397 (1989). | MR 953224 | Zbl 0697.14015

[13] Schwartz (N.), Madden (J. J.).— Semi-algebraic Function Rings and Reflectors of Partially Ordered Rings, Lecture Notes in Mathematics 1712. | MR 1719673 | Zbl 0967.14038

[14] Stone (M.H.).— Topological representations of distributive lattices and Brouwerian logics, Casopis, Mat. Fys., Praha, 67, p. 1-25 (1937).

[15] Schwartz (N.), Tressl (M.).— Elementary properties of minimal and maximal points in Zariski spectra, J. Algebra, vol. 323, no. 3, p. 698-728 (2010). | MR 2574858