On two problems from "Hyperidentities and Clones"

A hyperidentity $E$ can be viewed as a statement in second order logic. When combined with a similarity type $\tau$, it can also be considered as a set of first order statements. Based on examples from "A small basis for hyperassociativity", which included hyperassociativity and $\tau=<2>$, it was conjectured that each first order theory so produced was finitely axiomatizable. Part of the analysis suggested further investigating the relatively free 2-generated semigroup satisfying one or both of the equations $xxyxxyz=xxyyz$ and $zyyxx=zyxxyxx$. At ICM 1994, the conjecture above was refuted, and a finite basis problem arose: Is it decidable which pairs $<E,\tau>$ give rise to finitely axiomatizable theories? This problem will be examined, and its connections to other fields (e.g. symbolic dynamics) will be reviewed. In doing so, we give partial solutions to problems 27 and 28 from "Hyperidentities and Clones"