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Modal operators and the formal dual of Birkhoff's completeness theorem

Published online by Cambridge University Press:  01 April 2003

STEVE AWODEY  and
JESSE HUGHES
STEVE AWODEY
Affiliation:
Department of Philosophy, Carnegie Mellon University, Pittsburgh, PA 15213.
JESSE HUGHES
Affiliation:
Department of Computer Science, University of Nijmegen, 6525 ED Nijmegen, The Netherlands.
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Abstract

We present the dual to Birkhoff's variety theorem in terms of predicates over the carrier of a cofree coalgebra (that is, in terms of ‘coequations’). We then discuss the dual to Birkhoff's completeness theorem, showing how closure under deductive rules dualises to yield two modal operators acting on coequations. We discuss the properties of these operators and show that they commute. We prove as our main result the invariance theorem, which is the formal dual of Birkhoff's completeness theorem.

Type
Research Article
Information
Mathematical Structures in Computer Science , Volume 13 , Issue 2 , April 2003 , pp. 233 - 258
Copyright
2003 Cambridge University Press

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Footnotes

This research is part of the Logic of Types and Computation project at Carnegie Mellon University under the direction of Dana Scott.