Elementary Amalgamation and Joint Embedding Property for Intermediate Logics
Department of Mathematics, Tarbiat Modarres University, P.O.Box 14115-175, Tehran, Iran; Institute for Studies in Theoretical Physics and Mathematics (IPM); e-mail: bagheri{at}modares.ac.ir
School of Mathematics, Amirkabir University of Technology, Tehran, Iran; Institute for Studies in Theoretical Physics and Mathematics (IPM), P.O.Box 19395-5746, Tehran, Iran; e-mail: pourmahd{at}ipm.ir
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Abstract |
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In this paper we study the elementary amalgamation property (AP) and the joint embedding property (JEP) for intermediate logics. We point out the class of Kripke structures with elementary embedding can be viewed within abstract elementary class framework. Following this approach, both elementary AP and JEP can be considered quite naturally for intermediate logics. The main method for our investigations is the extension of Morleyization method from classical model theory to Kripke model theory. The almost-classical logic and almost-classical models have been defined. After verifying that the class of almost-classical models forms an abstract elementary class, we, furthermore, prove that the almost-classical logic neither has the elementary AP nor JEP property. We finally give an example of a non-classical intermediate logic which extends the almost-classical logic and has both elementary AP and JEP.
MSC 2000: 03C95, 03C90, 03B55.
Key Words: Intermediate logic • Kripke model • abstract elementary class • elementary amalgamation property • elementary joint embedding property • Morleyization • ultraproduct • almost-classical logic
Received for publication 27 November 2006.
1This research was in part supported by a grant from IPM, No. 83030043.
2This research was in part supported by a grant from IPM, No. 83030113.