Abstract
In this chapter, the algebraic duality that exists between relational structures and complete and atomic Boolean algebras with operators is studied. Every relational structure corresponds to a uniquely determined dual complete and atomic Boolean algebra with operators, namely the complex algebra of the structure, and conversely, every complete and atomic Boolean algebra with operators corresponds to a uniquely determined dual relational structure, namely the atom structure of the algebra. The duality between structures and algebras carries with it a corresponding duality between morphisms: every bounded homomorphism between relational structures corresponds to a dual complete homomorphism between the dual algebras, and conversely. The duality between the morphisms implies other dualities as well. For example, every inner subuniverse of a relational structure corresponds to a complete ideal in the dual algebra, and vice versa. Every bounded congruence on a relational structure corresponds to a complete subuniverse of the dual algebra, and vice versa. The disjoint union of a system of relational structures corresponds to the direct product of the system of dual algebras, and vice versa.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Blackburn, P., de Rijke, M., Venema, Y.: Modal Logic. Cambridge Tracts in Theoretical Computer Science, vol. 53, xxii + 554 pp. Cambridge University Press, Cambridge (2001)
Chin, L.H., Tarski, A.: Distributive and modular laws in the arithmetic of relation algebras. University of California Publications in Mathematics, New Series, vol. 1, pp. 341–384. University of California Press, Berkeley/Los Angeles (1951)
Everett, C.J., Ulam, S.: Projective algebra I. Am. J. Math. 68, 77–88 (1946)
Feferman, S.: Persistent and invariant formulas for outer extensions. Compos. Math. 20, 29–52 (1968)
Frayne, T., Morel, A.C., Scott, D.S.: Reduced direct products. Fundamenta Mathematicae 51, 195–228 (1962)
Gehrke, M., Harding, J., Venema, Y.: MacNeille completions and canonical extensions. Trans. Am. Math. Soc. 358, 573–590 (2006)
Givant, S.: The Structure of Relation Algebras Generated by Relativizations. Contemporary Mathematics, vol. 156, xvi + 134 pp. American Mathematical Society, Providence (1994)
Givant, S., Halmos, P.: Introduction to Boolean Algebras. Undergraduate Texts in Mathematics, xiv + 574 pp. Springer, New York (2009)
Goldblatt, R.I.: Metamathematics of modal logic, parts I and II. Rep. Math. Log. 6 and 7, pp. 41–78 and 21–52 (1976) respectively. (Reprinted in: Goldblatt, R.: Mathematics of Modality. Lecture Notes, vol. 43, pp. 9–79. CSLI Publications, Stanford (1993))
Goldblatt, R.I.: Varieties of complex algebras. Ann. Pure Appl. Log. 44, 173–242 (1989)
Henkin, L., Tarski, A.: Cylindric algebras. In: Dilworth, R.P. (ed.) Lattice Theory. Proceedings of Symposia in Pure Mathematics, vol. 2, pp. 83–113. American Mathematical Society, Providence (1961)
Henkin, L., Monk, J.D., Tarski, A.: Cylindric Algebras, Part I. Studies in Logic and the Foundations of Mathematics, vol. 64, vi + 508 pp. North-Holland, Amsterdam (1971)
Jónsson, B.: A survey of Boolean algebras with operators. In: Rosenberg, I.G., Sabidussi, G. (eds.) Algebras and Orders, pp. 239–286. Kluwer Academic, Dordrecht (1993)
Jónsson, B., Tarski, A.: Direct Decompositions of Finite Algebraic Systems. Notre Dame Mathematical Lectures, vol. 54, vi + 64 pp. University of Notre Dame, Notre Dame (1947)
Jónsson, B., Tarski, A.: Boolean algebras with operators. Part I. Am. J. Math. 73, 891–939 (1951)
Jónsson, B., Tarski, A.: Boolean algebras with operators. Part II. Am. J. Math. 74, 127–162 (1952)
Łoś, J.: Quelques remarques, théorèmes et problèmes sur les classes définissables d’algèbres. In: Mathematical Interpretation of Formal Systems. Studies in Logic and the Foundations of Mathematics, pp. 98–113. North-Holland, Amsterdam (1955)
Monk, J.D.: On representable relation algebras. Mich. Math. J. 11, 207–210 (1964)
Monk, J.D.: Completions of Boolean algebras with operators. Math. Nachr. 46, 47–55 (1970)
Peirce, C.S.: Note B. The logic of relatives. In: Peirce, C.S. (ed.) Studies in Logic by Members of the Johns Hopkins University. Little, Brown, and Company, Boston, 1883, pp. 187–203. (Reprinted by John Benjamins Publishing Company, Amsterdam, 1983)
Pierce, R.S.: Compact Zero-Dimensional Metric Spaces of Finite Type. Memoirs of the American Mathematical Society, vol. 130, 64 pp. American Mathematical Society, Providence (1972)
Sain, I.: Strong amalgamation and epimorphisms of cylindric algebras and Boolean algebras with operators. Mathematical Institute of the Hungarian Academy of Sciences, 44 pp. Preprint 17/1982 (1982)
Schröder, E.: Vorlesungen über die Algebra der Logik (exakte Logik), vol. III, Algebra und Logik der Relative, part 1. B. G. Teubner, Leipzig, 1895. (Reprinted by Chelsea Publishing Company, New York, 1966)
Segerberg, K.: An Essay in Classical Modal Logic. Filosofiska Studier Utgivna av Filosofiska Föreningen Och Filosofiska Institutionen Vid Uppsala Universitet, vol. 13, 500 pp. University of Uppsala, Uppsala (1971)
Tarski, A.: On the calculus of relations. J. Symb. Log. 6, 73–89 (1941)
Thomason, S.K.: Categories of frames for modal logic. J. Symb. Log. 40, 439–442 (1975)
Venema, Y.: Algebras and coalgebras. In: van Benthem, J., Blackburn, P., Wolter, F. (eds.) Handbook of Modal Logic. Studies in Logic and Practical Reasoning, vol. 3, pp. 331–426. Elsevier, Amsterdam (2006)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Givant, S. (2014). Algebraic Duality. In: Duality Theories for Boolean Algebras with Operators. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-06743-8_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-06743-8_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-06742-1
Online ISBN: 978-3-319-06743-8
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)