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Products in almost $f$ -algebras

Karim Boulabiar — 2000

Commentationes Mathematicae Universitatis Carolinae

Let $A$ be a uniformly complete almost $f$ -algebra and a natural number $p \in {3, 4, \dots}$ . Then $Π_{p} (A) = {a_{1} \dots a_{p}; a_{k} \in A, k = 1, \dots, p}$ is a uniformly complete semiprime $f$ -algebra under the ordering and multiplication inherited from $A$ with $Σ_{p} (A) = {a^{p}; 0 \leq a \in A}$ as positive cone.

An evaluating characterization of homomorphisms

Karim Boulabiar — 2008

Archivum Mathematicum

This work provides an evaluating complete description of positive homomorphisms on an arbitrary algebra of real-valued functions.

Arens regularity of lattice-ordered rings

Karim Boulabiar; Jamel Jabeur — 2010

Annales de la faculté des sciences de Toulouse Mathématiques

This work discusses the problem of Arens regularity of a lattice-ordered ring. In this prospect, a counterexample is furnished to show that without extra conditions, a lattice-ordered ring need not be Arens regular. However, as shown in this paper, it turns out that any $f$ -ring in the sense of Birkhoff and Pierce is Arens regular. This result is then used and extended to the more general setting of almost $f$ -rings introduced again by Birkhoff.