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.
In the present paper we deal with generalized $MV$-algebras ($GMV$-algebras, in short) in the sense of Galatos and Tsinakis. According to a result of the mentioned authors, $GMV$-algebras can be obtained by a truncation construction from lattice ordered groups. We investigate direct summands and retract mappings of $GMV$-algebras. The relations between $GMV$-algebras and lattice ordered groups are essential for this investigation.
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].
We denote by $K$ the class of all cardinals; put $K^{\prime }= K \cup \lbrace \infty \rbrace $. Let $\mathcal C$ be a class of algebraic systems. A generalized cardinal property $f$ on $\mathcal C$ is defined to be a rule which assings to each $A \in \mathcal C$ an element $f A$ of $K^{\prime }$ such that, whenever $A_1, A_2 \in \mathcal C$ and $A_1 \simeq A_2$, then $f A_1 =f A_2$. In this paper we are interested mainly in the cases when (i) $\mathcal C$ is the class of all bounded lattices $B$ having more than one element, or (ii) $\mathcal C$ is a class of lattice ordered groups.
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).
In this paper we deal with the of an $MV$-algebra $\mathcal A$, where $\alpha $ and $\beta $ are nonzero cardinals. It is proved that if $\mathcal A$ is singular and $(\alpha,2)$-distributive, then it is . We show that if $\mathcal A$ is complete then it can be represented as a direct product of $MV$-algebras which are homogeneous with respect to higher degrees of distributivity.