Globals of unary algebras (Q2643002)

A biunary algebra is an algebra of type \((2,2)\). The global \(2^{\mathcal A}\) of an algebra \({\mathcal A}\) with base set \(A\) is an algebra of the same type as \({\mathcal A}\) with base set \(2^A\) and fundamental operations defined via the complex product. A class of algebras of the same type is called globally determined if \({\mathcal A},{\mathcal B}\in K\) and \(2^{\mathcal A}\cong 2^{\mathcal B}\) together imply \({\mathcal A}\cong{\mathcal B}\). It is proved that for \(n\in\{1,2,3,4,5,6,9\}\) the class of \(n\)-element binary algebras is globally determined, whereas for \(i\in\{7,8\}\) it is not.