Abstract
We compare the monadic second-order theory of an arbitrary linear ordering L with the theory of the family of subsets of L endowed with the operation on subsets obtained by lifting the \(\max \) operation on L. We show that the two theories define the same relations. The same result holds when lifting the \(\min \) operation or both \(\max \) and \(\min \) operations.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Partially supported by TARMAC ANR agreement 12 BS02 007 01.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Büchi, J.R.: On a decision problem in restricted second-order arithmetic. In: Proceedings of the 1960 International Congress for Logic, Methodology and Philosophy, pp. 1–11. Stanford Univ. Press (1962)
Büchi, J.R.: The Monadic Second-Order Theory of All Countable Ordinals. Lecture Notes in Math., vol. 328. Springer, Heidelberg (1973)
Büchi, J.R., Zaiontz, C.: Deterministic automata and the monadic theory of ordinals \(\omega _2\). Zeitschrift für math. Logik und Grundlagen der Mat. 29, 313–336 (1983)
Choffrut, C., Grigorieff, S.: Logical theory of the additive monoid of subsets of natural integers. In: Adamatzky, A. (ed.) Automata, Universality, Computation. ECC, vol. 12, pp. 39–74. Springer, Heidelberg (2015)
Choffrut, C., Grigorieff, S.: Logical theory of the monoid of languages over a nontally alphabet. Fundamenta Informaticae 138(1–2), 159–177 (2015)
Gurevich, Y.: Elementary properties of ordered abelian groups. Algebra Logic 3(1), 5–39 (1964). English version in AMS Translations 46, 165–192 (1965)
Gurevich, Y., Magidor, M., Shelah, S.: The monadic theory of \(\omega _2\). J. Symbolic Logic 48(2), 387–398 (1983)
Gurevich, Y., Shelah, S.: Modest theory of short chains II. J. Symbolic Logic 44, 491–502 (1979)
Gurevich, Y., Shelah, S.: Monadic theory of order and topology in ZFC. Ann. Math. Logic 23, 179–198 (1982)
Gurevich, Y., Shelah, S.: Interpreting second-order logic in the monadic theory of order. J. Symbolic Logic 48, 816–828 (1983)
Gurevich, Y., Shelah, S.: The monadic theory and the ‘next world’. Israel J. Math. 49, 55–68 (1984)
Lifsches, S., Shelah, S.: The monadic theory of \(\omega _2\) may be complicated. Arch. Math. Logic 31, 207–213 (1992)
Rosenstein, J.G.: Linear Orderings. Academic Press, New York (1982)
Shelah, S.: The monadic theory of order. Ann. Math. 102, 379–419 (1975)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Choffrut, C., Grigorieff, S. (2015). Monadic Theory of a Linear Order Versus the Theory of its Subsets with the Lifted Min/Max Operations. In: Beklemishev, L., Blass, A., Dershowitz, N., Finkbeiner, B., Schulte, W. (eds) Fields of Logic and Computation II. Lecture Notes in Computer Science(), vol 9300. Springer, Cham. https://doi.org/10.1007/978-3-319-23534-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-319-23534-9_6
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-23533-2
Online ISBN: 978-3-319-23534-9
eBook Packages: Computer ScienceComputer Science (R0)