# Choice principles and lift lemmas

Document Type : Research Paper

Author

Faculty for Mathematics and Physics, IAZD, Leibniz Universit"at, Welfengarten 1, D 30167 Hannover, Germany.

Abstract

We show that in ${\bf ZF}$ set theory without choice, the Ultrafilter Principle (${\bf UP}$) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasicontinuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from ${\bf UP}$ but also from ${\bf DC}$, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ${\bf ZF}$ set theory.

Keywords

#### References

[1] Alexandrof, P., Diskrete Raume, Mat. Sb. (N.S.) 2 (1937), 501-518.
[2] Banaschewski, B., Coherent frames, in: B. Banaschewski and R.-E. Hofmann (eds.), Continuous Lattices, Lecture Notes in Math. 871, Springer, Berlin, 1981, 1-11.
[3] Banaschewski, B., The power of the ultrafilter theorem, J. London Math. Soc. 27(2) (1983), 193-202.
[4] Banaschewski, B., Prime elements from prime ideals, Order 2 (1985), 211-213.
[5] Banaschewski, B. and Erne, M., On Krull's separation lemma, Order 10 (1993), 253-260.
[6] Banaschewski, B. and Harting, R., Lattice aspects of radical ideals and choice principles, Proc. London Math. Soc. 50 (1985), 385-404.
[7] Birkhof, G., Lattice Theory", Amer. Math. Soc. Coll. Publ. 25, Providence, R.I., 1st ed. 1948, 3d ed. 1973.
[8] Brunner, N., Sequential compactness and the axiom of choice, Notre Dame J. Form. Log. 24 (1983), 89-92.
[9] Erne, M., Einfuhrung in die Ordnungstheorie", B.I.-Wissenschaftsverlag, Bibliographisches Institut, Mannheim, 1982.
[10] Erne, M., Order, Topology and Closure", University of Hannover, 1982.
[11] Erne, M., On the existence of decompositions in lattices, Algebra Universalis 16 (1983), 338-343.
[12] Erne, M., A strong version of the Prime Element Theorem, Preprint, University of Hannover, 1986.
[13] Erne, M., Ordnungs- und Verbandstheorie", Fernuniversitat Hagen, 1987.
[14] Erne, M., The ABC of order and topology, in: H. Herrlich and H.-E. Porst (eds.), Category Theory at Work", Heldermann, Berlin, 1991, 57-83.
[15] Erne, M., Prime ideal theorems and systems of finite character, Comment. Math. Univ. Carolinae 38 (1997), 513-536.
[16] Erne, M., Prime ideal theory for general algebras, Appl. Categ. Structures 8 (2000), 115-144.
[17] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445-489.
[18] Erne, M., Choiceless, pointless, but not useless: dualities for preframes, Appl. Categ. Structures 15 (2007), 541-572.
[19] Erne, M., Infinite distributive laws versus local connectedness and compactness properties, Topology Appl. 156 (2009) 2054-2069.
[20] Erne, M., The strength of prime ideal separation, sobriety, and compactness theorems, Preprint, Leibniz University Hannover, 2016. See also: Erne, M., Sober spaces, well-filtration and compactness principles, http://www.iazd.uni-hannover.de/ erne/preprints/sober.pdf (2007).
[21] Felscher, W., Naive Mengen und abstrakte Zahlen" III, B.I. Wissenschaftsverlag, Mannheim, 1979.
[22] Frasse, R., Theory of Relations", Studies in Logic 118, North-Holland, Amsterdam, 1986.
[23] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., Continuous Lattices and Domains", Oxford University Press, 2003.
[24] Gierz, G., Lawson, J.D., Stralka, A.R., Quasicontinuous posets, Houston J. Math. 9 (1983), 191-208.
[25] Heckmann, R. and Keimel, K., Quasicontinuous domains and the Smyth powerdomain, Electron. Notes Theor. Comput. Sci. 298 (2013), 215-232.
[26] Herrlich, H., Axiom of Choice", Lecture Notes in Math. 1876, Springer, Berlin Heidelberg, 2006.
[27] Hoft, H. and Howard, P., Well-ordered subsets of linearly ordered sets, Notre Dame J. Form. Log. 35 (1994), 413-425.
[28] Howard, P. and Rubin, J.E., Consequences of the Axiom of Choice", AMS Mathematical Surveys and Monographs 59, Providence, 1998.
[29] Isbell, J.R., Function spaces and adjoints, Math. Scand. 36 (1975), 317-339.
[30] Jech, T.J., The Axiom of Choice", North-Holland, Amsterdam, 1973.
[31] Johnstone, P.T., Scott is not always sober, in: B. Banaschewski and R.-E. Hofmann (eds.), Continuous Lattices", Lecture Notes in Math. 871, Springer, Berlin, 1981, 282-283.
[32] Johnstone, P.T., Stone Spaces", Cambridge University Press, 1982.
[33] Jung, A., Cartesian Closed Categories of Domains, CWI Tracts 66, Centrum voor Wiskunde en Informatica, Amsterdam (1989), 107 pp.
[34] Konig, D.,  Uber eine Schlussweise aus dem Endlichen ins Unendliche: Punktmengen. Kartenfarben. Verwandtschaftsbeziehungen. Schachspiel, Acta Lit. Sci. Reg. Univ. Hung. 3 (1927), 121-130.
[35] Krom, M., Equivalents of a weak axiom of choice, Notre Dame J. Form. Log. 22 (1981), 283-285.
[36] Moore, G.H., Zermelo's Axiom of Choice", Springer, Berlin Heidelberg NewYork, 1982.
[37] Picado, J. and Pultr, A., Frames and Locales", Birkhauser, Basel, 2012.
[38] Rubin, H., and Scott, D.S., Some topological theorems equivalent to the prime ideal theorem, Bull. Amer. Math. Soc. 60 (1954), 389 (Abstract).
[39] Rudin, M., Directed sets which converge, in: McAuley, L.F., and Rao, M.M. (eds.), General Topology and Modern Analysis", University of California, Riverside, 1980, Academic Press, 1981, 305-307.
[40] Scott, D.S., Prime ideal theorems for rings, lattices and Boolean algebras, Bull. Amer. Math. Soc. 60 (1954), 390 (Abstract).
[41] Tarski, A., Prime ideal theorems for Boolean algebras and the axiom of choice, Bull. Amer. Math. Soc. 60 (1954), 390-391 (Abstract).
[42] Tarski, A., Algebraic and axiomatic aspects of two theorems on sums of cardinals, Fund. Math. 35 (1948), 79-104.
[43] Wyler, O., Dedekind complete posets and Scott topologies, in: B. Banaschewski and R.E. Hofmann (eds.), Continuous Lattices", Lecture Notes in Math. 871, pringer, Berlin, 1981, 384-389.