Ideals of functions with compact support in the integer-valued case
-
Themba Dube
,
Oghenetega Ighedo
Abstract
For a zero-dimensional Hausdorff space X, denote, as usual, by C(X, ℤ) the ring of continuous integer-valued functions on X. If f ∈ C(X, ℤ), denote by Z(f) the set of all points of X that are mapped to 0 by f. The set
is the integer-valued analogue of the ideal of functions with compact support in C(X). By first working in the category of locales and then interpreting the results in spaces, we characterize this ideal in several ways. Writing ζ X for the Banaschewski compactification of X, we also explore some properties of ideals of C(X, ℤ) associated with subspaces of ζ X analogously to how one associates, for any Tychonoff space Y, subsets of β Y with ideals of C(Y).
(Communicated by L’ubica Holá)
Acknowledgement
Thanks are due to the referee for helpful comments that culminated in an improved paper.
References
[1] Baboolal, D.—Picado, J.—Pillay, P.—Pultr, A.: Hewitt’s irresolvabilty and induced sublocales in spatial frames, Quaest. Math. 43 (2020), 1601–1612.10.2989/16073606.2019.1646832Search in Google Scholar
[2] Ball, R. N.—Walters-Wayland, J.: C- and C*-quotients in pointfree topology, Dissertationes Math. (Rozprawy Mat.) 412 (2002), 62 pp.10.4064/dm412-0-1Search in Google Scholar
[3] Banaschewski, B.: Universal zero-dimensional compactifications. Categorical topology and its relation to analysis, algebra and combinatorics, (Prague, 1988), pp. 257–269, World Sci. Publ., Teaneck, NJ, 1989.Search in Google Scholar
[4] Banaschewski, B.: The real numbers in pointfree topology, Textos de Matem ática Série B, No. 12, Departamento de Matem ática da Universidade de Coimbra, 1997.Search in Google Scholar
[5] Banaschewski, B.: On the function ring functor in pointfree topology, Appl. Categ. Structures 13 (2005), 305–328.10.1007/s10485-005-5795-7Search in Google Scholar
[6] Banaschewski, B.: A remark on the ring of integer-valued continuous functions in pointfree topology, Topology Appl. 272 (2020), Art. ID 106964.10.1016/j.topol.2019.106964Search in Google Scholar
[7] Banaschewski, B.—Hong, S. S.: Completeness properties of function rings in pointfree topology, Comment. Math. Univ. Carolin. 44 (2003), 245–259.Search in Google Scholar
[8] Banaschewski, B.—Sioen, M.: Ring ideals and the Stone-Čech compactification in pointfree topology, J. Pure Appl. Algebra 214 (2010), 2159–2164.10.1016/j.jpaa.2010.02.019Search in Google Scholar
[9] Dube, T.: Some algebraic characterizations of F-frames, Algebra Universalis 62 (2009), 273–288.10.1007/s00012-010-0054-7Search in Google Scholar
[10] Dube, T.: Some ring-theoretic properties of almost P-frames, Algebra Universalis 60 (2009), 145–162.10.1007/s00012-009-2093-5Search in Google Scholar
[11] Dube, T.: On the ideal of functions with compact support in pointfree function rings, Acta Math. Hung. 129 (2010), 205–226.10.1007/s10474-010-0024-8Search in Google Scholar
[12] Dube, T.: Concerning P-sublocales and diconnectivity, Appl. Categ. Structures 27 (2019), 365–38.10.1007/s10485-019-09559-9Search in Google Scholar
[13] Dube, T.: Characterizing realcompact locales via remainders, Georgian Math. J. 28 (2021), 59–72.10.1515/gmj-2019-2027Search in Google Scholar
[14] Dube, T.—Ighedo, O.—Tlharesakgosi, B.: On ideals of rings of continuous integer-valued functions on a frame, Bull. Belg. Math. Soc. Simon Stevin 28 (2021), 429–458.Search in Google Scholar
[15] Dube, T.—Stephen, D. N.: Mapping ideals to sublocales, Appl. Categ. Structures 29 (2021), 747–772.10.1007/s10485-021-09634-0Search in Google Scholar
[16] Ferreira, M. J.—Picado, J.—Pinto, S. M.: Remainders in pointfree topology, Topology Appl. 245 (2018), 21–45.10.1016/j.topol.2018.06.007Search in Google Scholar
[17] Gutierréz Garc ía, J.—Picado, J.—Pultr, A.: Notes on Point-free Real Functions and Sublocales. Categorical Methods in Algebra and Topology, Math. Texts 46, Univ. Coimbra, Coimbra, 2014. %% 167–200. Search in Google Scholar
[18] Gillman, L.—Jerison, M.: Rings of Continuous Functions, Van Nostrand, Princeton, 1960.10.1007/978-1-4615-7819-2Search in Google Scholar
[19] Harris, D.: The local compactness of υ X, Pacific J. Math. 50 (1974), 469–476.10.2140/pjm.1974.50.469Search in Google Scholar
[20] Johnson, D. G.—Mandelker, M.: Functions with pseudocompact support, Gen. Topology Appl. 3 (1971), 331–338.10.1016/0016-660X(73)90020-2Search in Google Scholar
[21] Johnstone, P. T.: Stone Spaces, Cambridge University Press, Cambridge (1982).Search in Google Scholar
[22] Picado, J.—Pultr, A.: Frames and Locales: Topology without Points, Frontiers in Mathematics, Springer, Basel, 2012.10.1007/978-3-0348-0154-6Search in Google Scholar
[23] Picado, J.—Pultr, A.—Tozzi, A.: Joins of closed sublocales, Houston J. Math. 45 (2019), 21–38.Search in Google Scholar
[24] Yang, T.: Rings of Integer-valued Functions in Pointfree Topology, MSc Dissertation, McMaster University, Canada, 2004.Search in Google Scholar
[25] Yang, T.: Tensor products of rings 𝔪𝔏 of zero-dimensional frames, Order 36 (2019), 669–682.10.1007/s11083-019-09487-2Search in Google Scholar
© 2022 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Prof. RNDr. Július korbaš, CSC. passed away
- Kalmbach measurability In d0-algebras
- Quantifiers on L-algebras
- Ideals of functions with compact support in the integer-valued case
- An algebraic study of the logic S5’(BL)
- Triangular numbers and generalized fibonacci polynomial
- A general matrix series inversion pair and associated polynomials
- Quantum ostrowski type inequalities for pre-invex functions
- Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
- Approximating families for lattice outer measures on unsharp quantum logics
- On a generalized Lamé-Navier system in ℝ3
- Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
- On some classical properties of normed spaces via generalized vector valued almost convergence
- Fan-Hemicontinuity for the gradient of the norm in Hilbert space
- Solvability of mixed problems for heat equations with two nonlocal conditions
- The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
- A note on set-star-K-Menger spaces
- A bivariate extension of the Omega distribution for two-dimensional proportional data
- Bernstein polynomials based iterative method for solving fractional integral equations
- The symmetric 4-Player gambler’s problem with unequal initial stakes
- Enveloping action: Convergence spaces
Articles in the same Issue
- Prof. RNDr. Július korbaš, CSC. passed away
- Kalmbach measurability In d0-algebras
- Quantifiers on L-algebras
- Ideals of functions with compact support in the integer-valued case
- An algebraic study of the logic S5’(BL)
- Triangular numbers and generalized fibonacci polynomial
- A general matrix series inversion pair and associated polynomials
- Quantum ostrowski type inequalities for pre-invex functions
- Hermite-Hadamard type inequalities for interval-valued fractional integrals with respect to another function
- Approximating families for lattice outer measures on unsharp quantum logics
- On a generalized Lamé-Navier system in ℝ3
- Coercive and noncoercive elliptic problems with variable exponent Laplacian under Robin boundary conditions
- On some classical properties of normed spaces via generalized vector valued almost convergence
- Fan-Hemicontinuity for the gradient of the norm in Hilbert space
- Solvability of mixed problems for heat equations with two nonlocal conditions
- The global harnack estimates for a nonlinear heat equation with potential under finsler-geometric flow
- A note on set-star-K-Menger spaces
- A bivariate extension of the Omega distribution for two-dimensional proportional data
- Bernstein polynomials based iterative method for solving fractional integral equations
- The symmetric 4-Player gambler’s problem with unequal initial stakes
- Enveloping action: Convergence spaces
