FIRST ORDER PRESERVATION THEOREMS IN TWO-SORTED LANGUAGES.pdf

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FIRST ORDER PRESERVATION THEOREMS IN TWO-SORTED LANGUAGES.pdf

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FIRST ORDER PRESERVATION THEOREMS IN TWO-SORTED LANGUAGES

FIRST ORDER PRESERVATION THEOREMS IN TWO-SORTED LANGUAGES PHILIP OLINf The purpose of this note is to extend to two-sorted first order languages several well-known one-sorted preservation theorems, and to give some preservation theorems which are new even in the one-sorted case. For reasons of uniformity and economy we adopt the presentation of Lindstrom [5]. However the techniques of Keisler [2], [3] were used to find the theorems. It also seems likely that the methods of Makkai [7] could be used here and, in addition, to extend these results to various infinitary languages. The generalization of the work here to many-sorted languages is clear. Feferman [1] obtained such a result for sentences preserved by extensions. In section 1 we give the generalization of Lindstroms main theorem to two-sorted languages. In section 2 we apply this to get the extensions of some known preservation theorems in Theorem 2 .1 , and other preservation theorems in Theorems 2.2, 2.3 and 2.4. As applications of these, let S be the two-sorted theory of modules (see [8], [9]). A set A x of sentences is syntactically described such that a sentence 0 is preserved by homomorphisms of modules, where the homomorphism is an isomorphism of the base rings, if and only if there is a 9 in Ax such that S h (j -0 (Theorem 2.1). Let nx n2 ?3 ... be an arbitrary but fixed sequence of positive integers. Let 0 W M denote the countable weak direct power, and M the direct power, of the module M. Then F is a restricted isomorphism from ?WM onto ?aN if it is an isomorphism and, for all /, F(M 1 ) — N (F being an isomorphism of the base rings). A set A2 of sentences is syntactically described such that a sentence $ is preserved, from M to JV, by restricted isomorphisms from ?w M onto ?a N if and only if there is a 9 in A2 such that S h 4 - 9 (Theorem 2.2). This theorem in the one-sorted case would give a corresponding result with, for example, S the theory of groups. Other consequences, involving modules be