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Interpretability of the Cantor varieties. (English. Russian original) Zbl 0849.08007
Algebra Logic 34, No. 4, 258-262 (1995); translation from Algebra Logika 34, No. 4, 464-471 (1995).
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.
Reviewer: J.Duda (Brno)
##### MSC:
 08B05 Equational logic, Mal’tsev conditions 03C05 Equational classes, universal algebra in model theory
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##### References:
 [1] O. C. Garcia and W. Taylor, ?The lattice of interpretability types of varieties,?Mem. AMS,50, No. 305 (1984). · Zbl 0559.08003 [2] R. McKenzie and S. Swerczkowski, ?Non-covering in the interpretability of equational theories,?Alg. Univ.,30, No. 2, 157-170 (1993). · Zbl 0802.08004 · doi:10.1007/BF01196089 [3] D. M. Smirnov,Varieties of Algebras [in Russian], Nauka, Novosibirsk (1992). · Zbl 0778.08003 [4] D. M. Smirnov, ?Interpretability types of varieties and strong Mal’tsev conditions,?Sib. Mat. Zh.,35, No. 3, 683-695 (1994). · Zbl 0855.34086 · doi:10.1007/BF02106611
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