The distinguished completion $E(G)$ of a lattice ordered group $G$ was investigated by Ball [1], [2], [3]. An analogous notion for $MV$-algebras was dealt with by the author [7]. In the present paper we prove that if a lattice ordered group $G$ is a direct product of lattice ordered groups $G_i$ $(i\in I)$, then $E(G)$ is a direct product of the lattice ordered groups $E(G_i)$. From this we obtain a generalization of a result of Ball [3].
In this paper we deal with a homogeneity condition for an $MV$-algebra concerning a generalized cardinal property. As an application, we consider the homogeneity with respect to $\alpha $-completeness, where $\alpha $ runs over the class of all infinite cardinals.
In this paper, we study on the direct product of uninorms on bounded lattices. Also, we define an order induced by uninorms which are a direct product of two uninorms on bounded lattices and properties of introduced order are deeply investigated. Moreover, we obtain some results concerning orders induced by uninorms acting on the unit interval [0,1].
In this paper we deal with weakly homogeneous direct factors of lattice ordered groups. The main result concerns the case when the lattice ordered groups under consideration are archimedean, projectable and conditionally orthogonally complete.