A note on HSI-algebras and counterexamples to Wilkie's identity (Q1272248)

The ``High School Identities'' (briefly HSI) consist of the set of identities of unitary semiring together with the exponentiation laws. Tarski's ``High School Problem'' is whether the HSI form a basis for the equational theory of \(N=(N;+,\cdot ,\uparrow ,1)\). \textit{R. Gurevič} proved in 1985 the first counterexample with a rather complicated 59-element algebra. \textit{S. Burris} and \textit{S. Lee} show in 1992 that the smallest counterexample has its cardinality bounds in the interval \(\langle 7,15\rangle\). The author improves this estimation to the interval \(\langle 8,14\rangle\).