Translator Disclaimer
2019 The inversion characterizations of $C(\mathcal {L})$ for a locale $\mathcal {L}$
Richard N. Ball, Anthony W. Hager
Rocky Mountain J. Math. 49(7): 2107-2120 (2019). DOI: 10.1216/RMJ-2019-49-7-2107

Abstract

The category $\mathbf {W}$ is comprised of archimedean $\ell $-groups $G$ with distinguished weak unit and unit-preserving $\ell $-group homomorphisms. For $G \in \mathbf {W}$ there is the canonical Yosida representation $G \leq D(\mathcal {Y}G)$ as extended-real valued functions, with the unit $e_G$ represented as the constant $1$ function. We define $a \in G$ to be kernel-maximal if the $\mathbf {W}$-kernel generated by $a$ in $G$ is all of $G$, and we define $a$ to be Yosida invertible if there is some $b \in G$ with $ab = e_G$ in the partial multiplication inherited from $D(\mathcal {Y}G)$. The main theorem is that $G$ is isomorphic to $C(\mathcal {L})$ for some (identifiable) locale $\mathcal {L}$ iff $G$ is divisible, uniformly complete, and every kernel-maximal element of $G$ is Yosida invertible.

Citation

Download Citation

Richard N. Ball. Anthony W. Hager. "The inversion characterizations of $C(\mathcal {L})$ for a locale $\mathcal {L}$." Rocky Mountain J. Math. 49 (7) 2107 - 2120, 2019. https://doi.org/10.1216/RMJ-2019-49-7-2107

Information

Published: 2019
First available in Project Euclid: 8 December 2019

MathSciNet: MR4039960
Digital Object Identifier: 10.1216/RMJ-2019-49-7-2107

Subjects:
Primary: 06D22
Secondary: 06F20, 46A40, 46E05, 54B35, ‎54C30

Rights: Copyright © 2019 Rocky Mountain Mathematics Consortium

JOURNAL ARTICLE
14 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.49 • No. 7 • 2019
Back to Top