TITLE: Is HSP(A) = ISP(A) decidable?
QUESTION [8 upvotes]: Let $A$ be a finite algebra for some finitary signature.


Is it decidable whether $\mathbb{H}\mathbb{S}\mathbb{P}(A) = \mathbb{I}\mathbb{S}\mathbb{P}(A)$?


That is, whether the variety generated by $A$ equals the quasivariety generated by $A$?

REPLY [2 votes]: The paper www.math.uwaterloo.ca/~snburris/htdocs/MYWORKS/PAPERS/Reducts.pdf may be relevant.  It proves that given a finite set of quasiidentities defining a quasivariety generated by a finite algebra $A$ you can decide if the quasivariety is a variety. Other related results are there.