Preservation and Interpolation Through Binary Relations Between Theories

Hébert, Michel

doi:10.1002/malq.19890350206

Cited by 6 publications

(1 citation statement)

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“…(The proof of Makkai, which is quite complicated, lies on a much refined version of Theorem 2 also found in [2]). This has interesting consequences on the strength of the preservation theorem for direct products (see [3]). …”

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confidence: 99%

On the Commutativity of Ultraproducts with Direct Products

Hébert¹

1991

Proceedings of the American Mathematical Society

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Abstract.We show that an ultraproduct of direct products of structures is elementarily equivalent to a direct product (naturally defined over an ultraproduct of sets!) of ultraproducts of these structures.It is a well-known fact of model theory that "finite products commute with ultraproducts": if J is a finite set, D is an ultrafilter on a set /, and {11-, 1i e /, j £ J) is a set of structures, then there is a canonical isomorphism:(where no93; is the D-ultraproduct of the 93, 's and Flfê. is the direct product of the 23 's; a more precise notation would be nD(UjeJíi¡ . ; i £ I) = nj€J(FlD(U.i , ; j € /)), but the context prevents any ambiguity). This can be checked directly (the isomorphism being the obviously defined one), or seen as a consequence of a very general algebraic property; namely, that "filtered colimits commute with finite limits" in a wide class of "natural" categories.One readily finds counterexamples for infinite sets J . In fact, the commutativity cannot be extended to infinite J even if the isomorphism is replaced by an elementary equivalence: let I = J = N, and consider a statement 4>, which is true in a direct product F\.K?èk if and only if it is true in each = Wx(R(x)) for some relation symbol R ), and a filter D on I containing all the cofinite subsets; then, if one chooses the structures 11-such that il( ■ |= cj> if and only if i>j, one gets U^U^ ß (=

. '"' However, it seems to have been unnoticed before that there is a natural way to generalize to infinite sets J the elementary equivalence above. One has to take on the right hand side the ultrapower (of sets) FlDJ instead of J (this really generalizes the finite case, as the natural embedding J -► UDJ an isomorphism when J is finite): more explicitly, for each x £ FljJ = / , denote by [x]D its equivalence class in UDJ ; then we will see that _ rVnA,,) = nMDenDy(nDn. W.

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confidence: 99%