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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on order complete $ f$-algebras


Author: Boris Lavrič
Journal: Proc. Amer. Math. Soc. 100 (1987), 414-418
MSC: Primary 06F25; Secondary 54C10
DOI: https://doi.org/10.1090/S0002-9939-1987-0891137-5
MathSciNet review: 891137
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Abstract: Let $ A$ be an Archimedean uniformly complete $ f$-algebra with unit element. Then the following conditions are equivalent

(i) $ A$ is order complete.

(ii) Every regular algebra ideal in $ A$ is an order ideal.

(iii) Every finitely generated regular algebra ideal in $ A$ is a principal algebra ideal.

The proof is based on the fact that the range of every injective orthomorphism in an order complete Riesz space is an order ideal.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1987-0891137-5
Keywords: Riesz space, $ f$-algebra, orthomorphism
Article copyright: © Copyright 1987 American Mathematical Society

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