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 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.