Compactifications of partial frames via strongly regular ideals
-
John Frith
and
Anneliese Schauerte
Abstract
Partial frames provide a fertile context in which to do pointfree structured and unstructured topology, using a small collection of axioms of an elementary nature. Amongst other things they can be used to investigate similarities and differences between frames, σ-frames and κ-frames.
In this paper, the theory of strong inclusions for partial frames is used to describe compactifications of completely regular partial frames; the elements of these compactifications are given explicitly as strongly regular ideals. This is independent of and encompasses the theory of compactifications for frames. As an application, we revisit the Samuel compactification of a uniform partial frame.
References
[1] Adámek, J.—Herrlich H.—Strecker, G.: Abstract and Concrete Categories, John Wiley & Sons Inc., New York, 1990.Search in Google Scholar
[2] Baboolal, D.—Ori, R. G.: Samuel Compactification and Uniform Coreflection of Nearness Frames, Proceedings Symposium on Categorical Topology (1994), University of Cape Town, 1999.Search in Google Scholar
[3] Banaschewski, B.: Compactification of frames, Math. Nachr. 149 (1990), 105–116.10.1002/mana.19901490107Search in Google Scholar
[4] Banaschewski, B.: σ-frames, unpublished manuscript, 1980. Available at http://mathcs.chapman.edu/CECAT/members/Banaschewski_publicationsSearch in Google Scholar
[5] Banaschewski, B.—Gilmour, C. R. A.: Stone-Čech compactification and dimension theory for regular σ-frames, J. London Math. Soc. 39 (1989), 1–8.10.1112/jlms/s2-39.1.1Search in Google Scholar
[6] Banaschewski, B.—Gilmour, C. R. A.: Realcompactness and the cozero part of a frame, Appl. Categ. Struct. 9 (2001), 395–417.10.1023/A:1011225712426Search in Google Scholar
[7] Banaschewski, B.—Pultr, A.: Samuel compactification and completion of uniform frames, Math. Proc. Camb. Phil. Soc. 108 (1990), 63–78.10.1017/S030500410006895XSearch in Google Scholar
[8] Ebrahimi, M. M.—Tabatabaee, M. V.: Compactification of κ-frames, Bull. Iranian Math. Soc. 29 (2003), 55–64.Search in Google Scholar
[9] Frith, J.—Schauerte, A.: The Samuel compactification for quasi-uniform biframes, Topology Appl. 156 (2009), 2116–2122.10.1016/j.topol.2009.03.034Search in Google Scholar
[10] Frith, J.—Schauerte, A.: Uniformities and covering properties for partial frames (I), Categ. General Alg. Struct. Appl. 2 (2014), 1–21.Search in Google Scholar
[11] Frith, J.—Schauerte, A.: Uniformities and covering properties for partial frames (II), Categ. General Alg. Struct. Appl. 2 (2014), 23–35.Search in Google Scholar
[12] Frith, J.—Schauerte, A.: The Stone-Čech compactification of a partial frame via ideals and cozero elements, Quaest. Math. 39 (2016), 115–134.10.2989/16073606.2015.1023866Search in Google Scholar
[13] Frith, J.—Schauerte, A.: Completions of uniform partial frames, Acta Math. Hungar. 147 (2015), 116–134.10.1007/s10474-015-0514-9Search in Google Scholar
[14] Frith, J.—Schauerte, A.: Coverages give free constructions for partial frames, Appl. Categ. Structures 25 (2017), 303–321.10.1007/s10485-015-9417-8Search in Google Scholar
[15] Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge, 1982.Search in Google Scholar
[16] Mac Lane, S.: Categories for the Working Mathematician, Springer-Verlag, Heidelberg, 1971.10.1007/978-1-4612-9839-7Search in Google Scholar
[17] Madden, J. J.: κ-frames, J. Pure Appl. Algebra 70 (1991), 107–127.10.1016/0022-4049(91)90011-PSearch in Google Scholar
[18] Naidoo, I.: On Completion in the Category SSNσFRM, Math. Slovaca 63 (2013), 201–214.10.2478/s12175-012-0093-ySearch in Google Scholar
[19] Paseka, J.: Covers in generalized frames, In: General Algebra and Ordered Sets (Horni Lipova 1994), Palacky Univ. Olomouc, Olomouc pp. 84–99.Search in Google Scholar
[20] Picado, J.—Pultr, A.: Frames and Locales, Springer, Basel, 2012.10.1007/978-3-0348-0154-6Search in Google Scholar
[21] Samuel, P.: Ultrafilters and compactification of uniform spaces, Trans. Amer. Math. Soc. 64 (1948), 100–132.10.1090/S0002-9947-1948-0025717-6Search in Google Scholar
[22] Walters, J. L.: Compactifications and uniformities on σ-frames, Comment. Math. Univ. Carolinae 32 (1991), 189–198.Search in Google Scholar
[23] Walters, J. L.: Uniform Sigma Frames and the Cozero Part of Uniform Frames, Masters Dissertation, University of Cape Town, 1990.Search in Google Scholar
[24] Zenk, E. R.: Categories of partial frames, Algebra Universalis 54 (2005), 213–235.10.1007/s00012-005-1939-8Search in Google Scholar
[25] Zhao, D.: Nuclei on Z-Frames, Soochow J. Math. 22 (1996), 59–74.Search in Google Scholar
[26] Zhao, D.: On Projective Z-frames, Canad. Math. Bull. 40 (1997), 39–46.10.4153/CMB-1997-004-4Search in Google Scholar
© 2018 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- On the family of functions with closure of graphs in the Mendez ideals
- The predicate completion of a partial information system
- On lattices of z-ideals of function rings
- Compactifications of partial frames via strongly regular ideals
- Lifting components in clean abelian ℓ-groups
- Invariance of nonatomic measures on effect algebras
- On the alexander dual of path ideals of cycle posets
- Generalized multiplicative derivations in 3-prime near rings
- Cleft extensions for quasi-entwining structures
- Groups with positive rank gradient and their actions
- Certain results on q-starlike and q-convex error functions
- Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate
- Positive periodic solutions for singular high-order neutral functional differential equations
- A special class of functional equations
- Matrix mappings and general bounded linear operators on the space bv
- Skew-symmetric operators and reflexivity
- Derivatives of Hadamard type in vector optimization
- Cardinal functions of the hyperspace of convergent sequences
- Suzuki-type of common fixed point theorems in fuzzy metric spaces
- Second hankel determinat for certain analytic functions satisfying subordinate condition
Articles in the same Issue
- On the family of functions with closure of graphs in the Mendez ideals
- The predicate completion of a partial information system
- On lattices of z-ideals of function rings
- Compactifications of partial frames via strongly regular ideals
- Lifting components in clean abelian ℓ-groups
- Invariance of nonatomic measures on effect algebras
- On the alexander dual of path ideals of cycle posets
- Generalized multiplicative derivations in 3-prime near rings
- Cleft extensions for quasi-entwining structures
- Groups with positive rank gradient and their actions
- Certain results on q-starlike and q-convex error functions
- Faber polynomial coefficient estimates for subclass of bi-univalent functions defined by quasi-subordinate
- Positive periodic solutions for singular high-order neutral functional differential equations
- A special class of functional equations
- Matrix mappings and general bounded linear operators on the space bv
- Skew-symmetric operators and reflexivity
- Derivatives of Hadamard type in vector optimization
- Cardinal functions of the hyperspace of convergent sequences
- Suzuki-type of common fixed point theorems in fuzzy metric spaces
- Second hankel determinat for certain analytic functions satisfying subordinate condition
