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Mathematica Slovaca

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Volume 68, Issue 2

Issues

Lifting components in clean abelian -groups

Karim Boulabiar
  • Department of Mathematics Faculty of Mathematical, Physical and Natural Sciences of Tunis Tunis-El Manar University, 2092-El Manar, Tunis, Tunisia
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/ Samir Smiti
  • Department of Mathematics Faculty of Mathematical, Physical and Natural Sciences of Tunis Tunis-El Manar University, 2092-El Manar, Tunis, Tunisia
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Published Online: 2018-03-31 | DOI: https://doi.org/10.1515/ms-2017-0101

Abstract

Let G be an abelian -group with a strong order unit u > 0. We call G u-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.

Keywords: clean; component; f-algebra; ℓ-group; ℓ-ideal; lifting; step function; totally disconnected; uniform density; vector lattice

MSC 2010: Primary 06F20; 06F25; Secondary 46E25; 54A10

This research is supported by Research Laboratory LATAO Grant LR11ES12.

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About the article


Received: 2015-12-23

Accepted: 2016-08-14

Published Online: 2018-03-31

Published in Print: 2018-04-25


Citation Information: Mathematica Slovaca, Volume 68, Issue 2, Pages 299–310, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.1515/ms-2017-0101.

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