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Relations Old and New

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Relational Methods for Computer Science Applications

Part of the book series: Studies in Fuzziness and Soft Computing ((STUDFUZZ,volume 65))

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

$$ c\left( {\sigma {\kern 1pt} \rho } \right)a \Leftrightarrow {\exists _{b \in B}}\left( {c\sigma b \vee b\rho a} \right) $$

for all a ∈ A and c ∈ C, and ρ ∨: A ↛ B by

$$ a{\rho ^ \vee }b \Leftrightarrow v\rho a $$

.

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.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, QC, H3A 2K6, Canada

    Joachim Lambek

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  1. Joachim Lambek
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Editor information

Editors and Affiliations

  1. Institute of Telecommunications, ul. Szachowa 1, 04-894, Warsaw, Poland

    Ewa Orłowska

  2. Institute of Informatics, University of Warsaw, ul. Banacha 2, 02-097, Warsaw, Poland

    Andrzej Szałas

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© 2001 Springer-Verlag Berlin Heidelberg

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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

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  • 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

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