Abstract
It is known in topos theory that the axiom of choice implies that the topos is Boolean. In this paper we want to prove and generalize this result in the context of allegories. In particular, we will show that partial identities do have complements in distributive allegories with relational sums and total splittings assuming the axiom of choice. Furthermore, we will discuss possible modifications of the assumptions used in that theorem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Bird, R., de Moor, O.: Algebra of Programming. Prentice-Hall, Englewood Cliffs (1997)
Brink, C., Kahl, W., Schmidt, G. (eds.): Relational Methods in Computer Science. Advances in Computer Science. Springer, Vienna (1997)
Diaconescu, R.: Axiom of Choice and Complementation. Proc. AMS 51, 176–178 (1975)
Freyd, P., Scedrov, A.: Categories, Allegories. North-Holland, Amsterdam (1990)
Goldblatt, R.: Topoi: The Categorical Analysis of Logic. Studies in Logic and the Foundation of Mathematics, vol. 98. Elsevier, Amsterdam (1984)
Grätzer, G.: General lattice theory, 2nd edn. Birkhäuser, Basel (2003)
Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides 43, vol. 1. Oxford University Press, Oxford (2002)
Johnstone, P.: Sketches of an Elephant: A Topos Theory Compendium. Oxford Logic Guides 44, vol. 2. Oxford University Press, Oxford (2002)
Maddux, R.: On the derivation of identities involving projection functions. In: Csirmaz, Gabbay, de Rijke (eds.) Logic Colloquium 1992, pp. 145–173. Center for the Study of Languages and Information Publications, Stanford (1995)
Schmidt, G., Ströhlein, T.: Relationen und Graphen. Springer, Heidelberg (1989); English version: Relations and Graphs. Discrete Mathematics for Computer Scientists. EATCS Monographs on Theoret. Comput. Sci. Springer, Heidelberg (1993)
Tarski, A., Givant, S.: A Formalization of Set Theory without Variables. Colloquium Publications, vol. 41. AMS (1987)
Winter, M.: Goguen Categories. A Categorical Approach to L-Fuzzy Relations. Trends in Logic, vol. 25. Springer, Heidelberg (2007)
Kawahara, Y., Winter, M.: Cardinal Addition in Distributive Allegories. Submitted to RelMiCS 11/AKA 6
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Winter, M. (2009). Complements in Distributive Allegories. In: Berghammer, R., Jaoua, A.M., Möller, B. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2009. Lecture Notes in Computer Science, vol 5827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04639-1_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-04639-1_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04638-4
Online ISBN: 978-3-642-04639-1
eBook Packages: Computer ScienceComputer Science (R0)