Let $M(X,𝒜)$ ($M^{*}(X,𝒜)$) be the $f$-ring of all (bounded) real-measurable functions on a $T$-measurable space $(X,𝒜)$, let $M_K(X,𝒜)$ be the family of all $f\in M(X,𝒜)$ such that $\mathrm{coz}\left(f\right)$ is compact, and let $M_{\infty }(X,𝒜)$ be all $f\in M(X,𝒜)$ that $\{x\in X:|f\left(x\right)|\ge \frac{1}{n}\}$ is compact for any $n\in \mathbb{N}$. We introduce realcompact subrings of $M(X,𝒜)$, we show that $M^{*}(X,𝒜)$ is a realcompact subring of $M(X,𝒜)$, and also $M(X,𝒜)$ is a realcompact if and only if $(X,𝒜)$ is a compact measurable space. For every nonzero real Riesz map $\varphi :M(X,𝒜)\to ℝ$, we prove that there is an element $x_0\in X$ such that $\varphi \left(f\right)=f\left(x_0\right)$ for every $f\in M(X,𝒜)$ if $(X,𝒜)$ is a compact measurable space. We confirm...

In this article, it will be shown that every $\ell$-subgroup of a Specker $\ell$-group has singular elements and that the class of $\ell$-groups that are $\ell$-subgroups of Specker $\ell$-group form a torsion class. Methods of adjoining units and bases to Specker $\ell$-groups are then studied with respect to the generalized Boolean algebra of singular elements, as is the strongly projectable hull of a Specker $\ell$-group.

Dans cet article, nous tentons de généraliser à d’autres situations l’isomorphisme de groupes topologiques qui existe entre le groupe ${\displaystyle ℝ/\mathbb{Z}}$ et le groupe unitaire $𝕌=\{z\in ℂ/|z|=1\}$.Nous montrons que cet isomorphisme existe algébriquement en toute généralité : pour tout corps algébriquement clos $C$ et toute involution $c$ de $C$ les groupes $𝕌(C,c)=\{z\in C/zc(z)=1\}$ et ${\displaystyle C^{\<c\>}/\mathbb{Z}}$ sont isomorphes. Nous donnons ensuite un exemple d’involution $c_0$ de $ℂ$ qui n’est pas conjuguée, dans le groupe $\text{Aut}\left(ℂ\right)$, à la conjugaison complexe et telle que $𝕌(ℂ,c_0)$ soit topologiquement isomorphe...

2000 Mathematics Subject Classification: 03E04, 12J15, 12J25.We consider a construction of fields with symmetric gaps that are not semi-η1. By this construction we give examples of fields with different asymmetric gaps.