  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"
