The algebraic face of minimality

Frank Wolter



Operators which map subsets of a given set to the set of their minimal elements with respect to some relation R form the basis of a semantic approach in non-monotonic logic, belief revision, conditional logic and updating. In this paper we investigate operators of this type from an algebraic viewpoint. A representation theorem is proved and various properties of the resulting algebras are investigated. It is shown that they behave quite differently from known algebras related to logics, e.g. modal algebras and Heyting algebras.

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Blok, W., “Varieties of interior algebras”, Dissertation, University of Amsterdam, 1976.

Blok, W., D. Pigozzi, Algebraizable Logics, Memoirs of the American Mathematical Society, vol. 77, AMS, 1989.

Blok,W., D. Pigozzi, “On the structure of varieties with equationally definable principal congruences”, Algebra Universalis 15 (1982): 195–227.

Chagrov, A. V., M. V. Zakharyaschev, Modal Logic, Oxford University Press, 1997.

Chang, C., H. Keisler, Model Theory, Amsterdam, 1973.

Fine, K., “Logics containing K4. Part II”, Journal of Symbolic Logic 50 (1985): 619–651.

Goldblatt, R., “Metamathematics of modal logic”, Reports on Mathematical Logic 6 (1976): 41–78, 7 (1976): 21–52.

Koppelberg, S., Handbook of Boolean Algebras, vol. 1, North-Holland, 1989.

Kracht, M., “An almost general splitting theorem for modal logic”, Studia Logica 49 (1990): 455–470.

Kraus, Lehmann and Magidor, “Nonmonotonic reasoning, preferential models and cumulative logics”, Artificial Intelligence 44 (1990): 167–207.

Henkin, Monk and Tarski, Cylindric Algebras. Part 1, Amsterdam, 1971.

Makinson, D., “Five faces of minimality”, Studia Logica 52 (1993): 339–379.

McKenzie R., “Equational bases and non-modular lattice varieties”, Transactions of the American Mathematical Society 174 (1972): 1–43.

Wolter, F., “The structure of lattices of subframe logics”, Annals of Pure and Applied Logic 86 (1977): 47–100.

ISSN: 1425-3305 (print version)

ISSN: 2300-9802 (electronic version)

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