Order Boundedness of Order Continuous Orthogonally Additive Operators
Abstract
We prove that, under suitable assumptions, the space of order continuous orthogonally additive operators between Riesz spaces is a Riesz space. As a consequence, we give a negative answer to an open problem posed by Mykhaylyuk and Popov concerning the order continuity of the modulus of an order continuous orthogonally additive operator. We also show that, under an appropriate condition on the domain, every order continuous orthogonally additive operator is order bounded, obtaining an orthogonally additive analogue of the corresponding result for linear operators. Moreover, we establish the order boundedness result in ZF for atomic Riesz spaces and in ZF with dependent choice (DC) for Dedekind complete Riesz spaces. Finally, we characterize when every orthogonally additive operator is an abstract Uryson operator.
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BibTeX
@misc{uyar2026order,
title = {{Order Boundedness of Order Continuous Orthogonally Additive Operators}},
author = {Ayşe Uyar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17846},
eprint = {2610.17846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17846},
}
RIS
TY - UNPB
TI - Order Boundedness of Order Continuous Orthogonally Additive Operators
AU - Ayşe Uyar
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17846
UR - https://arxiv.org/abs/2610.17846
AB - We prove that, under suitable assumptions, the space of order continuous orthogonally additive operators between Riesz spaces is a Riesz space. As a consequence, we give a negative answer to an open problem posed by Mykhaylyuk and Popov concerning the order continuity of the modulus of an order continuous orthogonally additive operator. We also show that, under an appropriate condition on the domain, every order continuous orthogonally additive operator is order bounded, obtaining an orthogonally additive analogue of the corresponding result for linear operators. Moreover, we establish the order boundedness result in ZF for atomic Riesz spaces and in ZF with dependent choice (DC) for Dedekind complete Riesz spaces. Finally, we characterize when every orthogonally additive operator is an abstract Uryson operator.
ER -