In  this paper we give a description of the lattice 
$\Lambda({\cal MT})$ of subvarieties of monadic Tarski 
algebras introduced in \cite{Itu-Mon2}, and prove that 
quasivarieties coincide with varieties. We also 
investigate some properties of the lattice $L({\cal MB})$ 
of quasivarieties of monadic Boolean algebras determined 
in \cite{Ada-Dzi}, showing the difference when a constant 
is added to the language of monadic Tarski algebras, and 
we give a quasiidentity for each member of $L({\cal MB})$.