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.
The notion of bounded commutative residuated $\ell $-monoid ($BCR$ $\ell $-monoid, in short) generalizes both the notions of $MV$-algebra and of $BL$-algebra. Let $\c A$ be a $BCR$ $\ell $-monoid; we denote by $\ell (\c A)$ the underlying lattice of $\c A$. In the present paper we show that each direct product decomposition of $\ell (\c A)$ determines a direct product decomposition of $\c A$. This yields that any two direct product decompositions of $\c A$ have isomorphic refinements. We consider also the relations between direct product decompositions of $\c A$ and states on $\c A$.
Let α be an infinite cardinal. Let Tα be the class of all lattices which are conditionally α-complete and infinitely distributive. We denote by T'α the class of all lattices X such that X is infinitely distributive, α-complete and has the least element. In this paper we deal with direct factors of lattices belonging to T α - As an application, we prove a result of Cantor-Bernstein type for lattices belonging to the class T'α.