In this paper, we first will show that for any space X and any Wallman sublattice $mathcal{A}$ of $mathcal{R}(X)$ with $Z(X)^{sharp}{subseteq}mathcal{A}$, (${Phi}^{-1}_{mathcal{A}}(X)$, ${Phi}_{mathcal{A}}$) is the minimal quasi-F cover of X if and only if (${Phi}^{-1}_{mathcal{A}}(X)$, ${Phi}_{mathcal{A}}$) is a quasi-F cover of X and $mathcal{A}{subseteq}mathcal{Q}_X$. Using this, if X is a locally weakly Lindel$ddot{o}$f space, the set {$mathcal{A}|mathcal{A}$ is a Wallman sublattice of $mathcal{R}(X)$ with $Z(X)^{sharp}{subseteq}mathcal{A}$ and ${Phi}^{-1}_{mathcal{A}}(X)$ is the minimal quasi-F cover of X}, when partially ordered by inclusion, has the minimal element $Z(X)^{sharp}$ and the maximal element $mathcal{Q}_X$. Finally, we will show that any Wallman sublattice $mathcal{A}$ of $mathcal{R}(X)$ with $Z(X)^{sharp}{subseteq}mathcal{A}{subseteq}mathcal{Q}_X$, ${Phi}_{mathcal{A}_X}:{Phi}^{-1}_{mathcal{A}}(X){ ightarrow}X$ is $z^{sharp}$-irreducible if and only if $mathcal{A}=mathcal{Q}_X$.