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Hyperformulas and Solid Algebraic Systems

K. Denecke and D. Phusanga
Studia Logica: An International Journal for Symbolic Logic
Vol. 90, No. 2 (Nov., 2008), pp. 263-286
Published by: Springer
Stable URL: http://www.jstor.org/stable/40269007
Page Count: 24
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Hyperformulas and Solid Algebraic Systems
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Abstract

Defining a composition operation on sets of formulas one obtains a manysorted algebra which satisfies the superassociative law and one more identity. This algebra is called the clone of formulas of the given type. The interpretations of formulas on an algebraic system of the same type form a many-sorted algebra with similar properties. The satisfaction of a formula by an algebraic system defines a Galois connection between classes of algebraic systems of the same type and collections of formulas. Hypersubstitutions are mappings sending pairs of operation symbols to pairs of terms of the corresponding arities and relation symbols to formulas of the same arities. Using hypersubstitutions we define hyperformulas. Satisfaction of a hyperformula by an algebraic system defines a second Galois connection between classes of algebraic systems of the same type and collections of formulas. A class of algebraic systems is said to be solid if every formula which is satisfied is also satisfied as a hyperformula. On the basis of these two Galois connections we construct a conjugate pair of additive closure operators and are able to characterize solid classes of lgebraic systems.

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