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First Order Preservation Theorems in Two‐Sorted Languages
Author(s) -
Olin Philip
Publication year - 1972
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s2-4.4.631
Subject(s) - library science , order (exchange) , citation , work (physics) , research council , operations research , computer science , history , media studies , sociology , mathematics , linguistics , engineering , mechanical engineering , philosophy , government (linguistics) , economics , finance
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 Lindstrom's 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") — 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 being direct factors of other modules, are noted in the final paragraph of this paper.