Schreier and Hall varieties of unary algebras

To every variety of unary algebras is put into correspondence a pair $(R,I)$, where $R$ is a monoid and $I$ is its left ideal satisfying the condition $\nu\lambda=\nu$ for all $\nu\in I$ and $\lambda\in R$. Properties of this pair are indicated which are equivalent to the original variety being a Schreier one (subalgebras of free algebras are free) or being a Hall variety (retracts of free algebras are free).