A relative interpolation theorem for infinitary universal Horn logic and its applications |
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Authors: | Alexej P Pynko |
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Institution: | (1) Department 100, V. M. Glushkov Institute of Cybernetics of National Academy of Sciences of Ukraine, Glushkov prosp. 40, Kiev, 03680, Ukraine |
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Abstract: | In this paper we deal with infinitary universal Horn logic both with and without equality. First, we obtain a relative Lyndon-style
interpolation theorem. Using this result, we prove a non-standard preservation theorem which contains, as a particular case,
a Lyndon-style theorem on surjective homomorphisms in its Makkai-style formulation. Another consequence of the preservation
theorem is a theorem on bimorphisms, which, in particular, provides a tool for immediate obtaining characterizations of infinitary universal Horn classes without
equality from those with equality. From the theorem on surjective homomorphisms we also derive a non-standard Beth-style preservation
theorem that yields a non-standard Beth-style definability theorem, according to which implicit definability of a relation
symbol in an infinitary universal Horn theory implies its explicit definability by a conjunction of atomic formulas. We also
apply our theorem on surjective homomorphisms, theorem on bimorphisms and definability theorem to algebraic logic for general
propositional logic. |
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Keywords: | 03C40 03C75 03B22 03G99 08C15 |
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