Arithmetical identities in a 2-element model of Tarski's system (Q2765385)

Tarski's high school identities system (HSI) consists of the following identities: \(x+ y= y+x\), \((x+ y)+ z= x+(y+ z)\), \(x\cdot 1= x\), \(x\cdot y= y\cdot x\), \((x\cdot y)\cdot z= x\cdot (y\cdot z)\), \(x\cdot (y+ z)= x\cdot y+ x\cdot z\), \(1^x= 1\), \(x^1= x\), \(x^{y+ z}= x^y\cdot x^z\), \((x\cdot y)^z= x^z\cdot y^z\), \((x^y)^z= x^{y\cdot z}\).NEWLINENEWLINENEWLINEThe question whether HSI is a basis of all the identities of \(\mathbb{N}\), known as Tarski's High School Problem, was answered negatively in 1980 by A. J. Wilkie. So it is important to find identities which are not true on the small models of HSI; this is the main scope of the present paper (to meet in fact arithmetical identities in a 2-element model of Tarski's system).NEWLINENEWLINEFor the entire collection see [Zbl 0972.00028].