Abstract: | We study properties of free algebras in the Cantor varieties Cm,n. A free algebra of rank r in Cm,n is denoted FC
m,n(r). We argue that the following hold: (1) any two Cm,n-free algebras FC
m,n(r) and FC
m,n(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, r≥m, and s≥m, are elementary equivalent;
(2) any two Cm,n-free algebras FC
m,n(r) and FC
m,n(s) of ranks r and s, where r and s are arbitrary (finite or infinite) cardinals, are universally equivalent, that is, share
one ∀-theory; (3) an elementary theory Th(FC
m,n(r)) for an arbitrary Cm,n-free algebra of (finite or infinite) rank r, treated in a signature Ω, is decidable; (4) an elementary theory Th(K) for an
arbitrary nonempty class of free algebras in Cm,n, treated in a signature Ω, is decidable.
Translated fromAlgebra i Logika, Vol. 38, No. 2, pp. 228–248, March–April, 1999. |