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 Vladikavkaz. Mat. Zh., 2019, Volume 21, Number 3, Pages 14–23 (Mi vmj696)

Lattice structure on bounded homomorphisms between topological lattice rings

O. Zabeti

University of Sistan and Baluchestan, P.O. Box 98135-674, Zahedan, Iran

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.

Key words: locally solid $\ell$-ring, bounded group homomorphism, lattice ordered ring.

DOI: https://doi.org/10.23671/VNC.2019.3.36457

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UDC: 517.98
MSC: 13J25, 06F30
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Citation: O. Zabeti, “Lattice structure on bounded homomorphisms between topological lattice rings”, Vladikavkaz. Mat. Zh., 21:3 (2019), 14–23

Citation in format AMSBIB
\Bibitem{Zab19} \by O.~Zabeti \paper Lattice structure on bounded homomorphisms between topological lattice rings \jour Vladikavkaz. Mat. Zh. \yr 2019 \vol 21 \issue 3 \pages 14--23 \mathnet{http://mi.mathnet.ru/vmj696} \crossref{https://doi.org/10.23671/VNC.2019.3.36457} \elib{https://elibrary.ru/item.asp?id=40874235} 

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This publication is cited in the following articles:
1. Zabeti O., “Am-Spaces From a Locally Solid Vector Lattice Point of View With Applications”, Bull. Iran Math. Soc.
2. O. Zabeti, “A few remarks on bounded homomorphisms acting on topological lattice groups and topological rings”, Filomat, 34:9 (2020), 2897–2905
3. O. Zabeti, “Topological lattice rings with the $AM$-property”, Vladikavk. matem. zhurn., 23:1 (2021), 20–31
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