Interpretability of the Cantor varieties (Q1918794)

A Cantor variety \(C_n\), \(n \geq 2\), is a variety of algebras with one \(n\)-ary functional symbol \(g\) and \(n\) unary functional symbols \(f_1, \dots, f_n\) satisfying the following identities: \(f_i(g(x_1,\dots, x_n)) = x_i\), \(1 \leq i \leq n\), \(g(f_1(x),\dots,f_n(x)) = x\). An SC-theory (or a Mal'tsev theory) of a variety \(V\) is the collection of all strong Mal'tsev conditions satisfied in \(V\). Theorem I. The SC-theory of the Cantor variety \(C_2\) has bases of any finite length \(\geq 1\). Theorem II. The dimension of every Cantor variety \(C_n\) is infinite.