Isomorphic but not Lower Base-Isomorphic Cylindric Set Algebras

It is examined in several papers when is it true the isomorphism of two cylindric set algebras is a base-isomorphism. Problem 1. For which M ∈ CAα it is true that if for i = 0 and i = 1 Di ∈ Caα, f i : M→Di is an isomorphisms, and base Di = Vi imply that there exists a bijection g : V0 → V1 such that g = f1 ◦ f −1 0 (for notations and notions see [3] and [5]). By the Löwenheim-Skolem theorem this implication is hardly ever true. An important generalization of the baseisomorphisms is the sub-base-isomorphism and the lower-base-isomorphism, the equivalence relation generated by the relation sub-base-isomorphism. For lower-base isomorphism the above problem reads as follows. Problem 2. For which M ∈ CAα it is true that if for i = 0 and i = 1 Di ∈ Csα f i :M → Di is an isomorphism then f1 ◦ f −1 0 : D0 → D1 is a lower-base-isomorphism. There are sufficient conditions onM or on the base of Di in Theorem 1.3.6 of [4], in [6] and in Proposition 3.4 (3) of [1]. By Corollary 1 of [7] we have the following sufficient condition for Problem 2. Theorem 3. If M∈ Lfα is countable generated, for every n ∈ ω NrnM is atomic and Di ∈ Cs α , fi ∈ Is(M,Di) (i = 0, 1) then f1 ◦ f −1 0 : D0 → D1 is a lower base-isomorphism. On Isomorphic but not Lower-Base-Isomorphic Cylindric Set Algebras 231 Concerning Theorem 3 the following problems arise. Problem 4. Can some of the conditions of Theorem 3 omitted? Especially Problem 4 (a). Is the condition ‘M is countable generated’ omittable i.e. does there exist an Lfα M such that for every n ∈ ω NrnM is atomic, and for i = 0 and i = 1 Di ∈ Cs α , f i ∈ Is(M,Di) such that f1 ◦ f −1 0 is not a lower base-isomorphism? Problem 4 (b). Is it possible with the above conditions that D0 and D1 are not lower base-isomorphic? (Cf. Proposition 3.5 (3), Problem 3 and Problem 4 of [1].) We prove that the answer to Problem 4 (b) (and hence for Problem 4 (a) is in affirmative as follows. Theorem 5. (See [2].) For every α ≥ ω, there exists an χ1-generated M ∈ Lf , such that for every n ∈ ω NrnM is atomic and for i = 0 and i = 1 there exists a Di ∈ Cs α and an fi ∈ Is(M,Di) such that D0 and D1 are not lower-base-isomorphic. Acknowledgements. We thank S. Shelah and M. Makkai for their much precious help.