Abstract
Dedekind categories and similar structures provide a suitable framework to reason about binary relations in an abstract setting. Arrow categories extend this theory by certain operations and axioms so that additional aspects of L-fuzzy relations become expressible. In particular, arrow categories allow to identify crisp relations among all relations. On the other hand, the new operations and axioms in arrow categories force the category to be uniform, i.e., to be within a particular subclass of Dedekind categories. As an extension, arrow categories inherit constructions from Dedekind categories such as the definition of relational sums and splittings. However, these constructions are usually modified in arrow categories by requiring that certain relations are additionally crisp. This additional crispness requirement and the fact that the category is uniform raises a general question about these constructions in arrow categories. When can we guarantee the existence of the construction with and without the additional requirement of crispness in the given arrow category or an extension thereof? This paper provides a complete answer to this complex question for the two constructions mentioned.
M. Winter—The author gratefully acknowledges support from the Natural Sciences and Engineering Research Council of Canada (283267).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Freyd, P., Scedrov, A.: Categories, Allegories. North-Holland Mathematical Library, vol. 39, North-Holland, Amsterdam (1990)
Kawahara, Y., Furusawa, H.: Crispness and Representation Theorems in Dedekind Categories. DOI-TR 143, Kyushu University (1997)
Schmidt, G., Ströhlein, T.: Relations and graphs. Discrete mathematics for computer scientists. EATCS Monographs on Theoretical Computer Science. Springer, Heidelberg (1993). https://doi.org/10.1007/978-3-642-77968-8
Schmidt G.: Relational Mathematics. Encyplopedia of Mathematics and Its Applications, vol. 132. Cambridge University Press, Cambridge (2011)
Winter M.: Strukturtheorie heterogener Relationenalgebren mit Anwendung auf Nichtdetermismus in Programmiersprachen. Dissertationsverlag NG Kopierladen GmbH, München (1998)
Winter, M.: A Pseudo representation theorem for various categories of relations. Theory Appl. Categories 7(2), 23–37 (2000)
Winter, M.: A new algebraic approach to \(L\)-fuzzy relations convenient to study crispness. INS Inf. Sci. 139, 233–252 (2001)
Winter M.: Goguen Categories - A Categorical Approach to \(L\)-Fuzzy Relations. Trends in Logic, vol. 25. Springer, Heidelberg (2007). https://doi.org/10.1007/978-1-4020-6164-6
Winter, M.: Arrow categories. Fuzzy Sets Syst. 160, 2893–2909 (2009)
Winter M.: Type-n arrow categories. In: Höfner P., Pous D., Struth G. (eds.): Relational and Algebraic Methods in Computer Science (RAMiCS 2017). LNCS, vol. 11194, pp. 307–322. Springer, Heidelberg (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Winter, M. (2021). Relational Sums and Splittings in Categories of L-fuzzy Relations. In: Fahrenberg, U., Gehrke, M., Santocanale, L., Winter, M. (eds) Relational and Algebraic Methods in Computer Science. RAMiCS 2021. Lecture Notes in Computer Science(), vol 13027. Springer, Cham. https://doi.org/10.1007/978-3-030-88701-8_26
Download citation
DOI: https://doi.org/10.1007/978-3-030-88701-8_26
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-88700-1
Online ISBN: 978-3-030-88701-8
eBook Packages: Computer ScienceComputer Science (R0)