In this paper it is proved that an abelian lattice ordered group which can be expressed as a nontrivial lexicographic product is never affine complete.
Let $\Delta $ and $H$ be a nonzero abelian linearly ordered group or a nonzero abelian lattice ordered group, respectively. In this paper we prove that the wreath product of $\Delta $ and $H$ fails to be affine complete.
A generalized $MV$-algebra $\mathcal A$ is called representable if it is a subdirect product of linearly ordered generalized $MV$-algebras. Let $S$ be the system of all congruence relations $\rho $ on $\mathcal A$ such that the quotient algebra $\mathcal A/\rho $ is representable. In the present paper we prove that the system $S$ has a least element.
This paper is a continuation of a previous author’s article; the result is now extended to the case when the lattice under consideration need not have the least element.
For a pseudo $MV$-algebra $\mathcal A$ we denote by $\ell (\mathcal A)$ the underlying lattice of $\mathcal A$. In the present paper we investigate the algebraic properties of maximal convex chains in $\ell (\mathcal A)$ containing the element 0. We generalize a result of Dvurečenskij and Pulmannová.