The author shows that the semigroup variety generated by the monoid \(A_0^1=\langle a,b:a^2=a,\;b^2=b,\;ba=0\rangle\cup\{1\}\) of order five is hereditarily finitely based and contains countably infinitely many subvarieties. The proof of this result is divided into several parts. In contrast, the monoid variety generated by \(A_0^1\) is shown to contain only eight subvarieties.