Topological Lattice Rings with the \(AM\)-Property

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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: 10.46698/a8913-4331-4311-d Topological Lattice Rings with the \(AM\)-Property Zabeti, O. Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 1. Abstract: Motivated by the recent definition of the \(AM\)-property in locally solid vector lattices [O. Zabeti, doi: 10.1007/s41980-020-00458-7], in this note, we try to investigate some counterparts of those results in the category of all locally solid lattice rings. In fact, we characterize locally solid lattice rings in which order bounded sets and bounded sets agree. Furthermore, with the aid of the \(AM\)-property, we find conditions under which order bounded group homomorphisms and different types of bounded group homomorphisms coincide. Moreover, we show that each class of bounded order bounded group homomorphisms on a locally solid lattice ring \(X\) has the Lebesgue or the Levi property if and only if so is \(X\). Keywords: locally solid lattice ring, bounded group homomorphism, \(AM\)-property, Levi property, Lebesgue property Language: English Download the full text For citation: Zabeti, O. Topological Lattice Rings with the \(AM\)-Property, Vladikavkaz Math. J., 2021, vol. 23, no. 1, pp. 20-31. DOI 10.46698/a8913-4331-4311-d + References 1. Hong, L. Locally Solid Topological Lattice-Ordered Groups, Archivum Mathematicum, 2015, vol. 51, no. 2, pp. 107-128. DOI: 10.5817/AM2015-2-107. 2. Johnson, D. G. A Structure Theory for a Class of Lattice-Ordered Rings, Acta Mathematica, 1960, vol. 104, no. 3-4, pp. 163-215. DOI: 10.1007/BF02546389. 3. Mirzavaziri, M. and Zabeti, O. Topological Rings of Bounded and Compact Group Homomorphisms on a Topological Ring, Journal of Advanced Research in Pure Mathematics, 2011, vol. 3, no. 2, pp. 100-106. DOI: 10.5373/jarpm.588.100410. 4. Zabeti O. Lattice Structure on Bounded Homomorphisms between Topological Lattice Rings, Vladikavkaz Mathematical Journal, 2019, vol. 21, no. 3, pp. 14-21. DOI 10.23671/VNC.2019.3.36457. 5. Zabeti, O. \(AM\)-Property in Locally Solid Vector Lattices and Applications, Bulletin of the Iranian Mathematical Society, 2020. DOI: 10.1007/s41980-020-00458-7. 6. Zabeti, O. A Few Remarks on Bounded Homomorphisms Acting on Topological Lattice Groups and Topological Rings, Filomat, 2020, vol. 34, no. 9, pp. 2897-2905. DOI: 10.2298/FIL2009897Z. 7. Zabeti, O. A Few Remarks on Boundedness in Topological Modules and Topological Groups, Hacettepe Journal of Mathematics and Statistics,2019, vol. 48, no. 2, pp. 420-426. DOI: 10.15672/HJMS.2017.524. 8. Husain, T. Introduction to Topological Groups, W. B. Saunders Company, 1966. 9. Kocinac, Lj. D. R. 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. 10. Aliprantis, C. D. and Burkinshaw, O. Positive Operators, 2nd ed., Springer, 2006. 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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