Prime Ideal Theorems and systems of finite character

Marcel Erné

Commentationes Mathematicae Universitatis Carolinae (1997)

  • Volume: 38, Issue: 3, page 513-536
  • ISSN: 0010-2628

Abstract

top
We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if S is a system of finite character then so is the system of all collections of finite subsets of ⋃ S meeting a common member of S ), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth’s Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.

How to cite

top

Erné, Marcel. "Prime Ideal Theorems and systems of finite character." Commentationes Mathematicae Universitatis Carolinae 38.3 (1997): 513-536. <http://eudml.org/doc/248104>.

@article{Erné1997,
abstract = {We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if $\text\{S\}$ is a system of finite character then so is the system of all collections of finite subsets of $\bigcup \text\{S\}$ meeting a common member of $\text\{S\}$), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth’s Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.},
author = {Erné, Marcel},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {axiom of choice; compact; consistent; prime ideal; system of finite character; subbase; axiom of choice; compactness; consistency; prime ideal; system of finite character; subbase},
language = {eng},
number = {3},
pages = {513-536},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Prime Ideal Theorems and systems of finite character},
url = {http://eudml.org/doc/248104},
volume = {38},
year = {1997},
}

TY - JOUR
AU - Erné, Marcel
TI - Prime Ideal Theorems and systems of finite character
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1997
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 38
IS - 3
SP - 513
EP - 536
AB - We study several choice principles for systems of finite character and prove their equivalence to the Prime Ideal Theorem in ZF set theory without Axiom of Choice, among them the Intersection Lemma (stating that if $\text{S}$ is a system of finite character then so is the system of all collections of finite subsets of $\bigcup \text{S}$ meeting a common member of $\text{S}$), the Finite Cutset Lemma (a finitary version of the Teichm“uller-Tukey Lemma), and various compactness theorems. Several implications between these statements remain valid in ZF even if the underlying set is fixed. Some fundamental algebraic and order-theoretical facts like the Artin-Schreier Theorem on the orderability of real fields, the Erdös-De Bruijn Theorem on the colorability of infinite graphs, and Dilworth’s Theorem on chain-decompositions for posets of finite width, are easy consequences of the Intersection Lemma or of the Finite Cutset Lemma.
LA - eng
KW - axiom of choice; compact; consistent; prime ideal; system of finite character; subbase; axiom of choice; compactness; consistency; prime ideal; system of finite character; subbase
UR - http://eudml.org/doc/248104
ER -

References

top
  1. Artin E., Schreier O., Algebraische Konstruktion reeller Körper, Abh. Math. Sem. Hamb. Univ. 5 (1926), 85-99. (1926) 
  2. Aubert K.E., Theory of x -ideals, Acta Math. 107 (1962), 1-52. (1962) Zbl0108.26002MR0148773
  3. Banaschewski B., The power of the ultrafilter theorem, J. London Math. Soc. (2) 27 (1983), 193-202. (1983) Zbl0523.03037MR0692524
  4. Banaschewski B., Prime elements from prime ideals, Order 2 (1985), 211-213. (1985) Zbl0576.06010MR0815866
  5. Banaschewski B., A new proof that ``Krull implies Zorn'', Mathematical Logic Quarterly 40 (1994), 478-480. (1994) Zbl0813.03032MR1301940
  6. Banaschewski B., Erné M., On Krull's separation lemma, Order 10 (1993), 253-260. (1993) Zbl0795.06005MR1267191
  7. Crawley P., Dilworth R.P., Algebraic Theory of Lattices, Prentice-Hall, N.J., 1973. Zbl0494.06001
  8. Davey B.A., Priestley H.A., Introduction to Lattices and Order, Cambridge University Press, 1990. Zbl1002.06001MR1058437
  9. de Bruijn N.G., Erdös P., A colour problem for infinite graphs and a problem in the theory of relations, Indag. Math. 13 (1951), 371-373. (1951) MR0046630
  10. Ebbinghaus H.-D., Flum J., Thomas W., Mathematical Logic, Springer-Verlag, New York, 1991. Zbl1139.03001MR1278260
  11. Engeler E., Eine Konstruktion von Modellerweiterungen, Z. Math. Logik Grundlagen Math. 5 (1959), 126-131. (1959) Zbl0087.00904MR0109124
  12. Erné M., Semidistributivity, prime ideals and the subbase lemma, Rend. Circ. Math. Palermo II -XLI (1992), 241-250. (1992) MR1196618
  13. Erné M., A primrose path from Krull to Zorn, Comment. Math. Univ. Carolinae 36 (1995), 123-126. (1995) MR1334420
  14. Erné M., Prime ideal theorems for universal algebras, Preprint Univ. Hannover, 1995. 
  15. Erné M., Gatzke H., Convergence and continuity in partially ordered sets and semilattices, in: R.-E. Hoffmann and K.H. Hofmann (eds.), Continuous lattices and their applications, Lecture Notes in Pure and Appl. Math. 101, Marcel Dekker Inc., New York-Basel, 1985, pp.9-40. MR0825993
  16. Frink O., Topology in lattices, Trans. Amer. Math. Soc. 51 (1942), 569-583. (1942) Zbl0061.39305MR0006496
  17. Fuchs L., Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. Zbl0137.02001MR0171864
  18. Gähler W., Grundstrukturen der Analysis I, Akademie-Verlag and Birkhäuser Verlag, Berlin-Basel, 1977. MR0519344
  19. Gierz G., Hofmann K.H., Keimel K., Lawson J.D., Mislove M., Scott D.S., A Compendium of Continuous Lattices, Springer-Verlag, Berlin-Heidelberg-New York, 1980. Zbl0452.06001MR0614752
  20. Grätzer G., General Lattice Theory, Birkhäuser, Basel, 1978. MR0504338
  21. Halpern J., The independence of the axiom of choice from the Boolean prime ideal theorem, Fund. Math. 55 (1964), 57-66. (1964) Zbl0151.01002MR0164891
  22. Halpern J., Lévy A., The Boolean prime ideal theorem does not imply the axiom of choice, in: D. Scott (ed.), Axiomatic set theory, Proc. Symp. Pure Math., Univ. of California, Los Angeles 13 (1), (1967), 83-124. MR0284328
  23. Halpern D., Howard P.E., Cardinals m such that 2 m = m , Bull. Amer. Math. Soc. 76 (1970), 487-490. (1970) Zbl0223.02055MR0268034
  24. Henkin L., The completeness of the first order functional calculus, J. Symbolic Logic 14 (1949), 159-166. (1949) Zbl0034.00602MR0033781
  25. Hodges W., Krull implies Zorn, J. London Math. Soc. 19 (1979), 285-287. (1979) Zbl0394.03045MR0533327
  26. Jech T., The Axiom of Choice, North-Holland, Amsterdam-New York, 1973. Zbl0259.02052MR0396271
  27. Johnstone P., Almost maximal ideals, Fund. Math. 123 (1984), 197-209. (1984) Zbl0552.06004MR0761975
  28. Kelley J.L., The Tychonoff product theorem implies the axiom of choice, Fund. Math. 37 (1950), 75-76. (1950) Zbl0039.28202MR0039982
  29. Klimovsky G., Zorn's theorem and the existence of maximal filters and ideals in distributive lattices, Rev. Un. Mat. Argentina 18 (1958), 160-164. (1958) MR0132707
  30. Läuchli H., Coloring infinite graphs and the Boolean prime ideal theorem, Israel J. Math. 9 (1971), 420-429. (1971) MR0288051
  31. Lévy A., Remarks on a paper by J. Mycielski, Acta Math. Acad. Sci. Hungar. 14 (1963), 125-130. (1963) MR0146088
  32. Los J., Ryll-Nardzewski C., 0n the application of Tychonoff's theorem in mathematical proofs, Fund. Math. 38 (1951), 233-237. (1951) MR0048795
  33. Los J., Ryll-Nardzewski C., Effectiveness of the representation theory for Boolean algebras, Fund Math. 41 (1954), 49-56. (1954) MR0065527
  34. Moore G.H., Zermelo's Axiom of Choice - its Origins, Development and Influence, Springer-Verlag, New York-Heidelberg-Berlin, 1982. Zbl0497.01005MR0679315
  35. Mycielski J., Some remarks and problems on the colouring of infinite graphs and the theorem of Kuratowski, Acta Math. Acad. Sci. Hung. 12 (1961), 125-129. (1961) MR0130686
  36. Parovičenko I.I., Topological equivalents of the Tihonov theorem, Dokl. Akad. Nauk SSSR 184 (1969), 38-39 Soviet Math. Dokl. 10 (1969), 33-34. (1969) MR0238266
  37. Rav Y., Variants of Rado's selection lemma and their applications, Math. Nachr. 79 (1977), 145-165. (1977) Zbl0359.02066MR0476530
  38. Rav Y., Semiprime ideals in general lattices, J. Pure and Appl. Algebra 56 (1989), 105-118. (1989) Zbl0665.06006MR0979666
  39. Rubin H., Rubin J.E., Equivalents of the Axiom of Choice, II, North-Holland, Amsterdam-New York-Oxford, 1985. MR0798475
  40. Rubin H., Scott D.S., Some topological theorems equivalent to the prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389 (Abstract). (1954) 
  41. Sageev G., An independence result concerning the axiom of choice, Ann. Math. Logic 8 (1975), 1-184. (1975) Zbl0306.02060MR0366668
  42. Scott D.S., Prime ideals for rings, lattices and Boolean algebras, Bull. Amer. Math. Soc. 60 (1954), 390 (Abstract). (1954) 
  43. Tarski A., Prime ideal theorems for Boolean algebras and the axiom of choice. Prime ideal theorems for set algebras and ordering principles. Prime ideal theorems for set algebras and the axiom of choice, Bull. Amer. Math. Soc. 60 (1954), 390-391 (Abstracts). (1954) 
  44. Teichmüller O., Braucht der Algebraiker das Auswahlaxiom?, Deutsche Math. 4 (1939), 567-577. (1939) MR0000212
  45. Tukey J.W., Convergence and uniformity in topology, Annals of Math. Studies 2, Princeton, 1940. Zbl0025.09102MR0002515
  46. van Benthem J.F.A.K., A set-theoretical equivalent of the prime ideal theorem for Boolean algebras, Fund. Math. 89 (1975), 151-153. (1975) Zbl0363.04010MR0382003

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.