On The Function Rings of Pointfree Topology

  • Banaschewski, Bernhard
  • Received : 2005.04.15
  • Published : 2008.06.30


The purpose of this note is to compare the rings of continuous functions, integer-valued or real-valued, in pointfree topology with those in classical topology. To this end, it first characterizes the Boolean frames (= complete Boolean algebras) whose function rings are isomorphic to a classical one and then employs this to exhibit a large class of frames for which the functions rings are not of this kind. An interesting feature of the considerations involved here is the use made of nonmeasurable cardinals. In addition, the integer-valued function rings for Boolean frames are described in terms of internal lattice-ordered ring properties.


ring of continuous functions in pointfree topology;Boolean frames and their $\sigma$-characters;nonmeasurable cardinals;order complete f-rings with singular unit


  1. B. Banaschewski, The Real Numbers in Pointfree Topology. Textos de Matematica Ser. B, vol. 12, Departamento de Matematica da Universidade de Coimbra, 1997.
  2. B. Banaschewski, On the Function Ring Functor in Pointfree Topology, Appl. Categ. Structures, 13(2005), 305-328.
  3. B. Banaschewski and S. S. Hong, Completeness properties of function rings in pointfree topology, Comment. Math. Univ. Carolinae, 44(2003), 245-259.
  4. A. L. Foster, Generalized "Boolean" theory of universal algebras II: Identities and subdirect sums of functionally complete algebras, Math. Z., 59(1953), 191-199.
  5. L. Gillman and M. Jerison, Rings of Continuous Functions. Van Nostrand Reinhold Company, New York 1960.
  6. A. W. Hager, Algebras of measurable functions, Duke Math. J., 38(1971), 21-27.
  7. P. T. Johnstone, Stone spaces. Cambridge stud. adv. math., vol. 3, Cambridge University Press, Cambridge, 1982.
  8. S. Vickers, Topology via Logic. Cambridge Tracts Theoret. Comp. Sci., vol. 5, Cambridge University Press, Cambridge, 1985.

Cited by

  1. Covering maximal ideals with minimal primes vol.74, pp.3-4, 2015,
  2. More ring-theoretic characterizations of P-frames vol.14, pp.05, 2015,