We prove that Wilkie's identity holds in those natural HSI-algebras where each element has finite decomposition into components.Further, we construct a bunch of HSI-algebras that satisfy all the identities of the set of positive integers N. Then, based on the constructed algebras, we prove that the identities of N hold in the HSI-algebra of finite posets when the value of each variable is a poset having an isolated point.

All arithmetical identities involving 1, addition, multiplication and exponentiation are valid in every 2‐element HSI‐algebra. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

All arithmetical identities involving 1, addition, multiplication and exponentiation will be true in a 2-element model of Tarski's system if a certain sequence of natural numbers is not bounded. That sequence can be bounded only if the set of Fermat's prime numbers is finite.Mathematics Subject Classification: 03H15, 03C05.

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