On Topologies Induced by Order Relations on Dual Numbers

Nurcan İlayda Kaya; Buşra Aktaş; Olgun Durmaz; Halit Gündoğan

doi:10.47000/tjmcs.1807698

On Topologies Induced by Order Relations on Dual Numbers

Abstract

Inequalities are often utilized in different areas of mathematics to establish theorems. The presence of inequalities plays a crucial role in the foundations of these branches. In this study, we examine the properties of order relations in the dual number system, inspired by the characteristics of order relations defined on real numbers. With the help of these relations, topologies have been constructed in the space $\mathbb{D}^n$, and the geometric modelings of some elements of these topologies have been carried out.

Keywords

References

Adams, C., The Knot Book: An Elementary Introduction to the Mathematical Theory of Knots, American Mathematical Society, 2004.
Akbar, A. Fisher, B., Khan, M., Common fixed point theorems in complex valued metric spaces, Numerical Functional Analysis and Optimization, 32(2011), 243–253.
Aktaş, B., Durmaz, O., Gündoğan, H., The inequalities on dual numbers and their topological structures, Turkish Journal of Mathematics, 47(2023), 1318-1334.
Beckenbach F.E., Bellman R., Inequalities, Gottingen, Berlin, Springer-Verlag, 1961.
Cambou, A. D., Menon, N., Three dimensional structure of a sheet crumpled into a ball, Proc. Natl. Acad. Sci. U.S.A., 108(2011), 14741–14745.
Carlsson, G., Topology and data, Bulletin (New Series) of the American Mathematical Society, 46(2009), 255–308.
Clifford, W.K., Preliminary sketch of biquaternions, Proc. London Math. Soc.,4 (1873), 381–395.
Craig, J. J., Introduction to Robotics: Mechanics and Control, 3rd Ed. Prentice-Hall, 2004.

Croom, F. H., Principles of Topology, Saunders College Publishing, 1989.
Dimentberg F.M., The Screw Calculus and Its Applications in Mechanics, Moscow, Russia: Foreign Technology Division, Wright-Patterson Air Force Base, 1965.
Durmaz O, Aktaş B, Keçilioğlu O., An overview to analyticity of dual functions, Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics, 71(2022): 1094-1119.
Eckman, E., Connect the Shapes Crochet Motifs: Creative Techniques for Joining Motfis of all Shapes, Storey Publishing, 2012.
Farber, M., Invitation to Topological Robotics, European Mathematical Society, 2008.
Fink A.M., An essay on the history of inequalities, Journal of Mathematical Analysis and Applications, 249(2000), 118-134.
Fischer I., Dual-Number Methods in Kinematics, Statics and Dynamics, New York, USA: CRC Press, 1998.
Frydryszak M.A., Dual numbers and supersymmetric mechanics, Czechoslovak Journal of Physics, 55(2005), 1409-1414.
Görgülü, A., Ozdamar E., A Manifold of dual numbers and dual Lie group, Journal of Sci.and Arts of Gazi unv, 1(1988), 28-36.
Halmaghi E., Liljedahl P. ,Inequalities in the history of mathematics: From Peculiarities to a Hard Discipline, GeoGebra International Journal of Romania, 4(2015), 43-56.
Hardy G.H., Littlewood J.E., Polya G., Inequalities, Cambridge, England: Cambridge University Press, 1934.
Herman J, Kucera R, Simsa J., Equations and Inequalities: Elementary Problems and Theorems in Algebra and Number Theory, New York, USA: Springer-Verlag, 2000.
Horak, M., Disentangling Topological puzzles by using knot theory, Mathematics Magazine, 79(2006), 368–375.
Kandasamy W.B.V., Smarandache F., Dual Numbers, Ohio, USA: ZIP Publishing, 2012.
Kotelnikov A.P., Screw Calculus and Some Applications to Geometry and Mechanics, Kazan, Russia: Annals of the Imperial University of Kazan, 1895.
Li S, Ge Q.J., Rational bezier line-symmetric motions, Journal of Mechanical Design, 127(2005), 222-226.
Mashaghi, S., Jadidi, T., Koenderink, G., Mashaghi, A., Lipid nanotechnology, Int. J. Mol. Sci., 14(2013), 4242–4282.
Messelmi F., Analysis of dual functions, Annual Review of Chaos Theory, Bifurcations and Dynamical Systems 4(2013), 37-54.
Mitrinovic S.D., Analytic Inequalities, New York, USA: Springer-Verlag, 1970.
Munkres J.R., Topology, NJ, USA: Prentice Hall, 2000.
Prakash, A., Kumar, P., Certain results on bicomplex topologies and their comparison, Glob. J. Sci. Front. Res. Math. Decis. Sci. 164(2016), 39–61.
Richeson, D., Euler’s Gem: The Polyhedron Formula and The Birth of Topology, Princeton University Press, 2008.
Savic, A., Bilgin, M., Ersoy, S., Paunovic, M. Topologies of bihyperbolic numbers, Mathematics 10(2022), 4224.
Stadler, B.M.R., Stadler, P.F., Wagner, G.P., Fontana, W., The Topology of the possible: Formal spaces underlying patterns of evolutionary change, Journal of Theoretical Biology, 213(2001), 241–274
Stephenson, C., Lyon, D., H¨ubler, A., Topological Properties of a Self Assembled Electrical Network via AB Initio Calculation, Scientific Reports 7, 2017.
Study E., Geometrie der Dynamen, Leipzig, Germany: B.G. Teubner, 1903.
Srivastava, R., Certain topological aspects of bicomplex space, Bull. Pure Appl. Math. 2(2008), 222–234.
Srivastava, R.K., Singh, S., Certain bicomplex dictionary order topologies, Int. J. Math. Sci. Eng. Appl. 4(2010), 245–258.
Veldkamp G.R., On the use of dual numbers, vectors and matrices in instantaneous spatial kinematics, Mechanism and Machine Theory, 11(1976), 141-156.
Vickers, S., Topology via Logic, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, 1996.
Wald R.M., A new type of gauge invariance for a collection of massles spin-2 fields. II. Geometrical Interpretation, Classical and Quantum Gravity, 5(1987), 1279-1316.
Yau, S., Nadis, S., The Shape of Inner Space, Basic Books, 2010.

Details

Primary Language

English

Subjects

Algebraic and Differential Geometry

Journal Section

Research Article

Authors

Nurcan İlayda Kaya
0000-0001-8966-5225
Türkiye

Buşra Aktaş
0000-0002-1285-7250
Türkiye

Olgun Durmaz ^*
0000-0002-0913-3307
Türkiye

Halit Gündoğan
0000-0002-5806-2090
Türkiye

Publication Date

June 30, 2026

Submission Date

October 21, 2025

Acceptance Date

February 1, 2026

Published in Issue

Year 2026 Volume: 18 Number: 2

DOI

https://doi.org/10.47000/tjmcs.1807698

IZ

https://izlik.org/JA82WM37NW

Cite

RIS / Bibtex

APA

Kaya, N. İ., Aktaş, B., Durmaz, O., & Gündoğan, H. (2026). On Topologies Induced by Order Relations on Dual Numbers. Turkish Journal of Mathematics and Computer Science, 18(2), 526-540. https://doi.org/10.47000/tjmcs.1807698

AMA

1.Kaya Nİ, Aktaş B, Durmaz O, Gündoğan H. On Topologies Induced by Order Relations on Dual Numbers. TJMCS. 2026;18(2):526-540. doi:10.47000/tjmcs.1807698

Chicago

Kaya, Nurcan İlayda, Buşra Aktaş, Olgun Durmaz, and Halit Gündoğan. 2026. “On Topologies Induced by Order Relations on Dual Numbers”. Turkish Journal of Mathematics and Computer Science 18 (2): 526-40. https://doi.org/10.47000/tjmcs.1807698.

EndNote

Kaya Nİ, Aktaş B, Durmaz O, Gündoğan H (June 1, 2026) On Topologies Induced by Order Relations on Dual Numbers. Turkish Journal of Mathematics and Computer Science 18 2 526–540.

IEEE

[1]N. İ. Kaya, B. Aktaş, O. Durmaz, and H. Gündoğan, “On Topologies Induced by Order Relations on Dual Numbers”, TJMCS, vol. 18, no. 2, pp. 526–540, June 2026, doi: 10.47000/tjmcs.1807698.

ISNAD

Kaya, Nurcan İlayda - Aktaş, Buşra - Durmaz, Olgun - Gündoğan, Halit. “On Topologies Induced by Order Relations on Dual Numbers”. Turkish Journal of Mathematics and Computer Science 18/2 (June 1, 2026): 526-540. https://doi.org/10.47000/tjmcs.1807698.

JAMA

1.Kaya Nİ, Aktaş B, Durmaz O, Gündoğan H. On Topologies Induced by Order Relations on Dual Numbers. TJMCS. 2026;18:526–540.

MLA

Kaya, Nurcan İlayda, et al. “On Topologies Induced by Order Relations on Dual Numbers”. Turkish Journal of Mathematics and Computer Science, vol. 18, no. 2, June 2026, pp. 526-40, doi:10.47000/tjmcs.1807698.

Vancouver

1.Nurcan İlayda Kaya, Buşra Aktaş, Olgun Durmaz, Halit Gündoğan. On Topologies Induced by Order Relations on Dual Numbers. TJMCS. 2026 Jun. 1;18(2):526-40. doi:10.47000/tjmcs.1807698