×

Strong De Morgan’s law and the spectrum of a commutative ring. (English) Zbl 0562.18005

The ”strong De Morgan’s law” in a topos is the assertion that the object of truth-values is (internally) linearly ordered. In Lect. Notes Math. 753, 479-491 (1979; Zbl 0445.03041), the reviewer showed that this condition holds in the topos of sheaves on a topological space X iff every closed suspace of X is extremally disconnected. The main result of this paper is that the condition holds for the spectrum of a Noetherian domain R iff R is a Dedekind domain (an example is given to show that the Noetherian condition cannot be dropped).
Reviewer: P.T.Johnstone

MSC:

18B25 Topoi
13F05 Dedekind, Prüfer, Krull and Mori rings and their generalizations
03G30 Categorical logic, topoi

Citations:

Zbl 0445.03041
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] {\scB. Banaschewski}, The power of the ultrafilter theorem, preprint. · Zbl 0523.03037
[2] Birkhoff, G, On the lattice theory of ideals, Bull. amer. math. soc., 40, 613-619, (1934) · JFM 60.0093.01
[3] Eilenberg, S; Kelly, G.M, Closed categories, () · Zbl 0192.10604
[4] Gierz, G; Hofmann, K.H; Keimel, K; Lawson, J.D; Mislove, M; Scott, D.S, A compendium of continuous lattices, (1980), Springer-Verlag Berlin/New York/Heidelberg · Zbl 0452.06001
[5] Grothendieck, A; Verdier, J.L, Théories des topos (SGA4), () · Zbl 0256.18008
[6] Johnstone, P.T, Conditions related to de Morgan’s law, (), 479-491 · Zbl 0445.03041
[7] Johnstone, P.T, Stone spaces, (1982), Cambridge Univ. Press Cambridge · Zbl 0499.54001
[8] Johnstone, P.T, The point of pointless topology, Bull. amer. math. soc., 8, 41-53, (1983) · Zbl 0499.54002
[9] Johnstone, P.T, Topos theory, () · Zbl 0368.18001
[10] Krull, W, Axiomatische begründung der allgemeinen idealtheorie, sitzungsberichte der physikalesch, Medicinischen societät zu erlangen 56, (1924) · JFM 50.0073.02
[11] Larsen, M; McCarthy, P, Multiplicative theory of ideals, () · Zbl 0237.13002
[12] Lawvere, F.W, Metric spaces, generalized logic, and closed categories, Rend. sem. mat. fis. milano, (1973)
[13] Macauley, F.S, The algebraic theory of modular systems, (1916), Cambridge Univ. Press New York/London
[14] MacDonald, I.G, Algebraic geometry: an introduction to schemes, (1968), Benjamin New York
[15] Rosenthal, K.I, Rigidity in closed categories and generalized sup-inf theorems, J. pure appl. algebra, 24, 41-57, (1982) · Zbl 0485.18006
[16] Stone, M.H, Topological representation of distributive lattices and Brouwerian logics, Casopis pest. mat. fys, 67, 1-25, (1937) · Zbl 0018.00303
[17] Ward, M; Dilworth, R, Residuated lattices, Trans. amer. math. soc., 45, 335-354, (1939) · JFM 65.0084.01
[18] Ward, M, Structure residuation, Ann. of math, 39, 558-569, (1938) · JFM 64.0073.02
[19] Willard, S, General topology, (1970), Addison-Wesley Reading, Mass · Zbl 0205.26601
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.