It has been observed by Banaschewski [1], Proposition 1, that, in the notation of the author's paper [2], if P ⊆ FU is hereditary and finitely productive. This fact does not require the use of injectives as in [2].

In a recent note [1], Arkeryd presents a counterexample to my conjecture [2]. The same example appears in a preprint by von Grudzinski [3], who also points out that the corollary to Theorem 2 does not follow from the proof of the theorem as stated. The fact that x3 + xy3 and x3 + xy3y3 are right equivalent, but not under quasi-identity, provides a counterexample. The proof on p. 170 of [2] that is also incorrect. In order to establish this result, we note that, from equation (31), it suffices to show that ξ(0, t) = 0 for all t. But, from equation (30), f(ξ(0, t)) = 0. If now 0 is an isolated zero of f, we are done. If not, we may assume ξ(0, t) = 0 without loss of generality. The corollary to Theorem 4 shows that this latter case is, in fact, vacuous.