The embedding theorem for Cantor varieties

Let m and n be fixed integers, with 1 ≤ m ≤ n. A Cantor variety Cm,n is a variety of algebras with m n-ary and n m-ary basic operations which is defined in a signature Ω = {g1,...,gm,f1,...,fn} by the identities fi(g1(x1,...,xn),...,gm (x1,...,xn)) = xi, i = 1,...,n, gj(f1(x1,...,xm),...,fn (x1,...,xm)) = xj, j = 1,...,m. We prove the following: (a) every partial Cm,n-algebra A is isomorphically embeddable in the algebra G = 〈A; S (A)〉 of Cm,n; (b) for every finitely presented algebra G = 〈A; S〉 in Cm,n, the word problem is decidable; (c) for finitely presented algebras in Cm,n, the occurrence problem is decidable; (d) Cm,n has a hereditarily undecidable elementary theory. © 2001 Plenum Publishing Corporation.