The paper is devoted to the study of the ordered set $A\mathcal{K}\left(X,\alpha \right)$ of all, up to equivalence, $A$-compactifications of an Alexandroff space $\left(X,\alpha \right)$. The notion of $A$-weight (denoted by $aw\left(X,\alpha \right)$) of an Alexandroff space $\left(X,\alpha \right)$ is introduced and investigated. Using results in ([7]) and ([5]), lattice properties of $A\mathcal{K}\left(X,\alpha \right)$ and $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ are studied, where $A{\mathcal{K}}_{\alpha \mathcal{w}}\left(X,\alpha \right)$ is the set of all, up to equivalence, $A$-compactifications $Y$ of $\left(X,\alpha \right)$ for which $w\left(Y\right)=aw\left(X,\alpha \right)$. A characterization of the families of bounded functions generating an $A$-compactification of $\left(X,\alpha \right)$ is obtained. The notion...