Tarski's finite basis problem via $\mathbf {A}({\mathcal T})$

R. McKenzie has recently associated to each Turing machine ${\mathcal T}$ a finite algebra $\mathbf {A} ({\mathcal T})$ having some remarkable properties. We add to the list of properties, by proving that the equational theory of $\mathbf {A}({\mathcal T})$ is finitely axiomatizable if ${\mathcal T}$ halts on the empty input. This completes an alternate (and simpler) proof of McKenzie's negative answer to A. Tarski's finite basis problem. It also removes the possibility, raised by McKenzie, of using $\mathbf {A}({\mathcal T})$ to answer an old question of B. Jónsson.