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.
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 $\c A$.
In this paper we apply the notion of cell of a lattice for dealing with graph automorphisms of lattices (in connection with a problem proposed by G. Birkhoff).
Freytes proved a theorem of Cantor-Bernstein type for algbras; he applied certain sequences of central elements of bounded lattices. The aim of the present paper is to extend the mentioned result to the case when the lattices under consideration need not be bounded; instead of sequences of central elements we deal with sequences of internal direct factors of lattices.