Abstract
Defining a composition operation on sets of formulas one obtains a many-sorted 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 algebraic systems.
Similar content being viewed by others
References
Denecke, K., D. Lau, R. Pöschel, and D. Schweigert, Hyperidentities, Hyperequational classes, and clone congruences, Contributions to General Algebra 7, Verlag Hölder-Pichler-Tempsky, Wien 1991, pp. 97–118.
Denecke, K., and S. L. Wismath, Hyperidentities and Clones, Gordon and Breach Science Publishers, 2000.
Denecke, K., and S. L. Wismath, Universal Algebra and Applications in Theoretical Computer Science, Chapman and Hall/CRC, 2002.
Graczyńska E., Schweigert D.: ‘;Hyperidentities of a given type’. Algebra Universalis 27 (1990), 305–318
Koppitz, J., and K. Denecke, M-solid Varieties, Springer 2006.
Mal’cev A.I.: Algebraic Systems. Akademie-Verlag, Berlin 1973
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Denecke, K., Phusanga, D. Hyperformulas and Solid Algebraic Systems. Stud Logica 90, 263–286 (2008). https://doi.org/10.1007/s11225-008-9152-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11225-008-9152-3