Lattice Structure on Bounded Homomorphisms Between Topological Lattice Rings

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2019.3.36457 Lattice Structure on Bounded Homomorphisms Between Topological Lattice Rings Zabeti, O. Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 3. Abstract: Suppose \(X\) is a topological ring. It is known that there are three classes of bounded group homomorphisms on \(X\) whose topological structures make them again topological rings. First, we show that if \(X\) is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on \(X\). Now, assume that \(X\) is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on \(X\). Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable Riesz-Kantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups. Keywords: locally solid \(\ell\)-ring, bounded group homomorphism, lattice ordered ring. Language: English Download the full text For citation: Zabeti, O. Lattice Structure on Bounded Homomorphisms Between Topological Lattice Rings, Vladikavkaz Math. J., 2019, vol. 21, no. 3, pp. 14-23. DOI 10.23671/VNC.2019.3.36457 + References 1. Birkhoff, G. Lattice-Ordered Groups, Annals of Mathematics, 1942, vol. 43, no. 2, pp. 298-331. DOI: 10.2307/1968871. 2. Clifford, A. H. Partially Ordered Abelian Groups, Annals of Mathematics, 1940, vol. 41, no. 3, pp. 465-473. DOI: 10.2307/1968728. 3. Smarda, B. Topologies in \(l\)-Groups, Archivum Mathematicum (Brno), 1967, vol. 3, no. 2, pp. 69-81. 4. Smarda, B. Some Types of Topological \(l\)-Groups, Publ. Fac. Sci. Univ. J. E. Purkyne (Brno), 1969, vol. 507, pp. 341-352. 5. Hong, L. Locally Solid Topological Lattice-Ordered Groups, Archivum Mathematicum (Brno), 2015, vol. 51, no. 2, pp. 107-128. 6. Birkhoff, G. and Pierce, R. S. Lattice-Ordered Rings , Anais da Academia Brasileira de Ciencias, 1956, vol. 28, pp. 41-69. 7. Johnson, D. G. A Structure Theory for a Class of Lattice-Ordered Rings, Acta Mathematica, 1960, vol. 104, no. 3-4, pp. 163-215. 8. Warner, S. Topological Fields, North-Holland: Elsevier Sci. Publ., 1989, North-Holland Math. Stud., vol. 157, ch. 1-2. 9. Arnautov, V., Glavatsky, S. and Mikhalev, A. A. Introduction to the Theory of Topological Rings and Modules, Monographs and Textbooks in Pure and Appl. Math., New York, Marcel Dekker, 1996. 10. Steinberg, S. A. Lattice-Ordered Rings and Modules, New York, Springer-Verlag, 2010. DOI: 10.1007/978-1-4419-1721-8. 11. Mirzavaziri, M. and Zabeti, O. Topological Rings of bounded and Compact Group Homomorphisms on a Topological Ring, J. Adv. Res. Pure Math., 2011, vol. 3, no. 2, pp. 101-106. 12. Kocinac, Lj. and Zabeti, O. Topological Groups of Bounded Homomorphisms on a Topological Group, Filomat, 2016, vol. 30, no. 3, pp. 541-546. DOI: 10.2298/FIL1603541K. 13. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, 2nd edition, Springer, 2006. DOI: 10.1007/978-1-4020-5008-4. 14. Erkursun-Ozcan, N., Gezer, N. A. and Zabeti, O. Spaces of \(u\tau\)-Dunford-Pettis and \(u\tau\)-Compact Operators on Locally Solid Vector Lattices , Matematicki Vesnik , 2019 [in press]. 15. Zabeti, O. A Few Remarks on Boundedness in Topological Groups and Topological Modules, Hacettepe Journal of Mathematics and Statistics , 2019, vol. 48, no. 2, pp. 420-426. DOI: 10.15672/HJMS.2017.524. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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