Abstract
We note that (binary) relations on a set form a (partially) ordered monoid with involution, which is also residuated and complete, hence a quantale. Relations between sets form an ordered category with involution. If ρ: A ↛ B and σ: B ↛ C, one defines σρ: A ↛ C by
for all a ∈ A and c ∈ C, and ρ ∨: A ↛ B by
.
The Partial order between relations A ↛ B is defined elementwise.
We shall discuss some appearances of relations in anthropology, linguistics, computer science, algebra and category theory.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Andréka H. and Mikulás S. (1993) The completeness of the Lambek calculus for relational semantics. J. of Logic, Language and Information 3:1–37
Barr M., Grillet P.A. and van Osdal D.H. (1971) Exact categories and categories of sheaves. Springer LNM 236
Bhargava M. and Lambek J. (1983) A production grammar for Hindi kinship terminology. Theoretical Linguistics 10:227–245
Bhargava M. and Lambek J. (1995) A rewrite system for the Western Pacific. Theoretical Linguistics 21:241–253
Birkhoff G. (1967) Lattice Theory. Amer. Math. Soc. New York
Calenko M.S., Gisin V.B. and Raikov D.A. (1984) Ordered categories with involution. Dissertations Mathematicae (= Rozprawy Matematyczne) 227:1–11
Casadio C. (2000) Noncommutative linear logic in linguistics. Manuscript
Chomsky N. (1979) Language and responsibility. Pantheon Books. New York
Findlay G.D. (1960) Reflexive homomorphic relations. Canad. Math. Bull. 3:131–132
Freyd P. and Scedrov A. (1990) Categories and allegories. North Holland, Amsterdam
Goursat É. Sur les substitutions orthogonales. Ann. Sci. École Normale Supérieure (3) 6:9–102
Lambek J. (1957) Goursat’s theorem and the Zassenhaus lemma. Canad. J. Math. 10:45–56
Lambek J. (1958) The mathematics of sentence structure. Amer. Math. Monthly 65:154–169
Lambek J. (1964) Goursat’s theorem and homological algebra. Can. Math. Bull. 7:597–608
Lambek J. (1986) A production grammar for English kinship terminology. Theoretical Linguistics 13:19–36
Lambek J. (1996) The butterfly and the serpent. In: Agliano P. and Ursini A. (Eds.) Logic and Algebra, Marcel Dekkert, New York, 161–179
Lambek J. (1997) Relations in operational categories. J. Pure and Applied Algebra 116:221–248
Lambek J. (2000) Diagram chasing in ordered categories with involution. J. Pure and Applied Algebra, to appear
Lambek J. (2000) Relations: binary relations in the social and mathematical sciences. In: Cantini A., Casari E., Minari P. (Eds.) Logic in Florence, Kluwer Academic Publishers, to appear
Leach E. (1958) Concerning Trobriand clans and the kinship category tabu. In: Goody J. (Ed.) The development cycle of domestic groups. Cambridge Papers on Social Anthropology 1, Cambridge University Press
Longo G., Moggi E. (1991) Constructive natural deduction and its “w-set” interpretation. Math. Structures in Computer Science 1,215–254
Lounsbury F.G. (1965) Another view of Trobriand kinship categories. In: Hammel E.A. (Ed.) Formal Semantics, American Anthropologist 67 (5) Part 2 142–185
Malinowski B. (1932) Sexual life of savages. 3rd edition. Routledge and Kegan Paul Ltd., London
Mac Lane S. (1961) An algebra of additive relations. Proc. Nat. Acad. Sci. USA 47(7):1043–1051
McLarty C. (1992) Elementary categories, elementary toposes. Claderon Press. Oxford
Mitschke A. (1971) Implication algebras are 3-permutable and 3-distributive. Algebra Universalis 1:182–186
Riguet J. (1950) Quelques propriétés des relations difonctionelles. C.R. Acad. Sci. Paris. Sér. I Math. 230:1999–2000
van Benthem J. (1991) Language in action. Elsevier
Yetter D.N. (1990) Quantales and (non-commutative) linear logic. J. Symbolic Logic 55:41–64
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Lambek, J. (2001). Relations Old and New. In: Orłowska, E., Szałas, A. (eds) Relational Methods for Computer Science Applications. Studies in Fuzziness and Soft Computing, vol 65. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1828-4_8
Download citation
DOI: https://doi.org/10.1007/978-3-7908-1828-4_8
Publisher Name: Physica, Heidelberg
Print ISBN: 978-3-662-00362-6
Online ISBN: 978-3-7908-1828-4
eBook Packages: Springer Book Archive