Let $G$ be an Archimedean $\ell $-group. We denote by $G^d$ and $R_D(G)$ the divisible hull of $G$ and the distributive radical of $G$, respectively. In the present note we prove the relation $(R_D(G))^d=R_D(G^d)$. As an application, we show that if $G$ is Archimedean, then it is completely distributive if and only if it can be regularly embedded into a completely distributive vector lattice.
In this note we prove that there exists a Carathéodory vector lattice $V$ such that $V\cong V^3$ and $V\ncong V^2$. This yields that $V$ is a solution of the Schröder-Bernstein problem for Carathéodory vector lattices. We also show that no Carathéodory Banach lattice is a solution of the Schröder-Bernstein problem.
In this paper we deal with the notions of projectability, spliting property and Dedekind completeness of lattice ordered groups, and with the relations between these notions.
In this paper we deal with a pseudo effect algebra $\Cal A$ possessing a certain interpolation property. According to a result of Dvurečenskij and Vettterlein, $\Cal A$ can be represented as an interval of a unital partially ordered group $G$. We prove that $\Cal A$ is projectable (strongly projectable) if and only if $G$ is projectable (strongly projectable). An analogous result concerning weak homogeneity of $\Cal A$ and of $G$ is shown to be valid.
Let D be the system of all distributive lattices and let D0 be the system of all L ∈ D such that L possesses the least element. Further, let D1 be the system of all infinitely distributive lattices belonging to D0. In the present paper we investigate the radical classes of the systems D, D0 and D1.