an:00907282
Zbl 0849.08007
Smirnov, D. M.
Interpretability of the Cantor varieties
EN
Algebra Logic 34, No. 4, 258-262 (1995); translation from Algebra Logika 34, No. 4, 464-471 (1995).
0002-5232 1573-8302
1995
j
08B05 03C05
interpretability; basis; Cantor variety; SC-theory; Mal'tsev theory; strong Mal'tsev conditions; dimension
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.
J.Duda (Brno)