Weak $(\mathfrak{m},\;\mathfrak{n})$-distributivity of lattice ordered groups and of generalized MV-algebras

Abstract

Let $\mathfrak{m}\;{\text{and}}\;\mathfrak{n}$ be cardinals. The concept of weak $(\mathfrak{m},\;\mathfrak{n})$-distributivity of Boolean algebras was intoduced by Sikorski. In the present paper we investigate this concept for lattice ordered groups and for generalized MV-algebras. We prove that the collection of all lattice ordered groups which are weakly $(\mathfrak{m},\;\mathfrak{n})$-distributive is a radical class. An analogous result is valid for generalized MV-algebras. A Sikorski’s result concerning Boolean algebras is extended for the case of generalized MV-algebras.