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Lifting components in clean abelian -groups

  • Karim Boulabiar EMAIL logo and Samir Smiti
From the journal Mathematica Slovaca

Abstract

Let G be an abelian -group with a strong order unit u > 0. We call Gu-clean after Hager, Kimber, and McGovern if every element of G can be written as a sum of a strong order unit of G and a u-component of G. We prove that G is u-clean if and only if u-components of G can be lifted modulo any -ideal of G. Moreover, we introduce a notion of u-suitable -groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the -group G is u-clean when and only when it is u-suitable. Also, we show that if E is a vector lattice, then E is u-clean if and only if the space of all u-step functions of E is u-uniformly dense in E. As applications, we will generalize a result by Banaschewski on maximal -ideals of an archimedean bounded f-algebras to the non-archimedean case. We also extend a result by Miers on polynomially ideal C(X)-type algebras to the more general setting of bounded f-algebras.


This research is supported by Research Laboratory LATAO Grant LR11ES12.



Communicated by Vincenzo Marra


Acknowledgement

The authors would like to thank the referee for his comments, suggestions, and constructive criticism which improved considerably the content and the writing of this paper.

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Received: 2015-12-23
Accepted: 2016-8-14
Published Online: 2018-3-31
Published in Print: 2018-4-25

© 2018 Mathematical Institute Slovak Academy of Sciences

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