We denote by $K$ the class of all cardinals; put $K^{\text{'}}=K\cup \left\{\infty \right\}$. Let $𝒞$ be a class of algebraic systems. A generalized cardinal property $f$ on $𝒞$ is defined to be a rule which assings to each $A\in 𝒞$ an element $fA$ of $K^{\text{'}}$ such that, whenever $A_1,A_2\in 𝒞$ 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) $𝒞$ is the class of all bounded lattices $B$ having more than one element, or (ii) $𝒞$ 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).