Let $\alpha $ be an infinite cardinal. In this paper we define an interpolation rule $\mathop {\mathrm IR}(\alpha )$ for lattice ordered groups. We denote by $C (\alpha )$ the class of all lattice ordered groups satisfying $\mathop {\mathrm IR}(\alpha )$, and prove that $C (\alpha )$ is a radical class.
Let $\frak m$ be an infinite cardinal. We denote by $C_\frak m$ the collection of all $\frak m$-representable Boolean algebras. Further, let $C_\frak m^0$ be the collection of all generalized Boolean algebras $B$ such that for each $b\in B$, the interval $[0,b]$ of $B$ belongs to $C_\frak m$. In this paper we prove that $C_\frak m^0$ is a radical class of generalized Boolean algebras. Further, we investigate some related questions concerning lattice ordered groups and generalized $MV$-algebras.
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.
It is proved that a radical class $\sigma $ of lattice-ordered groups has exactly one cover if and only if it is an intersection of some $\sigma $-complement radical class and the big atom over $\sigma $.
In this paper we prove that the collection of all weakly distributive lattice ordered groups is a radical class and that it fails to be a torsion class.