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Categories of frames for modal logic1

Published online by Cambridge University Press:  12 March 2014

S. K. Thomason*
Affiliation:
Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6

Extract

§1. A complete atomic modal algebra (CAMA) is a complete atomic Boolean algebra with an additional completely additive unary operator. A (Kripke) frame is just a binary relation on a nonempty set. If is a frame, then is a CAMA, where mX = {y ∣ (∃x)(y < x Є X)}; and if is a CAMA then is a frame, where is the set of atoms of and b1 < b2b1mb2 ≠∅.

Now , and the validity of a modal formula on is equivalent to the satisfaction of a modal algebra polynomial identity by and conversely, so the validity-preserving constructions on frames ought to be in some sense equivalent to the identity-preserving constructions on CAMA's. The former are important for modal logic, and many of the results of universal algebra apply to the latter, so it is worthwhile to fix precisely the sense of the equivalence.

The most important identity-preserving constructions on CAMA's can be described in terms of homomorphisms and complete homomorphisms. Let and be the categories of CAMA's with homomorphisms and complete homomorphisms, respectively. We shall define categories and of frames with appropriate morphisms, and show them to be dual respectively to and . Then we shall consider certain identity-preserving constructions on CAMA's and attempt to describe the corresponding validity-preserving constructions on frames.

The proofs of duality involve some rather detailed calculations, which have been omitted. All the category theory a reader needs to know is in the first twenty pages of [7].

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1975

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Footnotes

1

This work was done while the author was on Sabbatical Leave at the University of Canterbury, Christchurch, and was supported in part by The Canada Council and the National Research Council of Canada.

References

REFERENCES

[1]Ésakia, L. L., Topological Kripke models, Doklady Akademii Nauk SSSR, vol. 214 (1974), pp. 298301. (Russian.)Google Scholar
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