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Lifting components in clean abelian $\ell$-groups

In: Mathematica Slovaca, vol. 68, no. 2
Karim Boulabiar - Samir Smiti
Detaily:
Rok, strany: 2018, 299 - 310
Kľúčové slová:
clean, component, $f$-algebra, $\ell$-group, $\ell$-ideal, lifting, step function, totally disconnected, uniform density, vector lattice
O článku:
Let $G$ be an abelian $\ell$-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 $\ell$-ideal of $G$. Moreover, we introduce a notion of $u$-suitable $\ell$-groups (as a natural analogue of the corresponding notion in Ring Theory) and we prove that the $\ell$-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 $\ell$-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.
Ako citovať:
ISO 690:
Boulabiar, K., Smiti, S. 2018. Lifting components in clean abelian $\ell$-groups. In Mathematica Slovaca, vol. 68, no.2, pp. 299-310. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0101

APA:
Boulabiar, K., Smiti, S. (2018). Lifting components in clean abelian $\ell$-groups. Mathematica Slovaca, 68(2), 299-310. 0139-9918. DOI: https://doi.org/10.1515/ms-2017-0101
O vydaní: