Majorants, Uniformities, and $bo-$Convergence in $L-$Scaled Spaces

Year 2025, Volume: 17 Issue: 2, 304 - 309, 30.12.2025

Niyazi Anil Gezer

https://doi.org/10.47000/tjmcs.1617617

Abstract

This article investigates the relations between uniform continuity and $bo$-convergence in $L$-scaled spaces, which are ultrametric spaces with distances measured using elements from a lattice $L$ with the smallest element. The $bo$-convergence in $L$-scaled spaces is defined by means of the order convergence in the lattice $L.$ We examine how monotone majorants can be employed to bound the behavior of functions and their iterates in these spaces. A central result demonstrates that if a function is uniformly continuous on an $L$-scaled space, then images of $bo$-convergent sequences have $bo$-convergent subsequences. Several results related to the uniform structure of $L$-scaled spaces and dominated operators are presented.

Keywords

Dominated map , ultrametric space , uniformity , lattice.

Ethical Statement

I agree with the ethical principles outlined in the "Publication Ethics and Publication Malpractice Statement," including the responsibilities of all parties involved in publishing, adherence to originality, avoidance of plagiarism, and commitment to ethical conduct in research and publication.

References

• Birkhoff, G., Lattice Theory, Publications of the AMS, 1940 (republished in 1967).
• Engelking, R., General Topology, 2nd Edition, Sigma Series in Pure Mathematics, 6, Heldermann Verlag, Berlin.
• Garg, V.K., Introduction to Lattice Theory with Computer Science Applications, John Wiley & Sons, 2015.
• Kusraev, A.G., Dominated Operators, Dordrecht, the Netherlands: Kluwer Academic Publishers, 2000.
• Ochsenius, H., Schikhof, W., Banach spaces over fields with an infinite rank valuation, In: p-adic Functional Analysis, Lect. Notes Pure Appl. Math., 207 (1999), 233–293, edited by J. Kakol, N. De Grande De Kimpe, and C. Perez-Garcia, Marcel Dekker.
• Priess-Crampe, S., Ribenboim, P., Generalized ultrametric spaces I, Abh. Math. Semin. Univ. Hambg., 66(1996), 55–73.
• Priess-Crampe, S., Ribenboim, P., Fixed points, combs and generalized power series, Abh. Math. Semin. Univ. Hambg., 63(1993), 227–244.
• Vulikh, B. Z., Introduction to the Theory of Partially Ordered Spaces, translated from the Russian by Leo F. Boron, with the editorial collaboration of A.C. Zaanen and K. Iseki, Wolters-Noordhoff Scientific Publications, Groningen, 1967.