Course on Algebra and Coalgebra

This is a back up reference of the page of the course on Algebra and Coalgebra (Autumn 2013).

Capita Selecta in

Modal Logic, Algebra and Coalgebra (Autumn 2013)

This page concerns the course `Capita Selecta in Modal Logic, Algebra and Coalgebra’, taught at the University of Amsterdam from October – December 2013. 

Contents of these pages


  • For the exam you need to study the material that you can find here (that is, the indicated sections of the Algebra|Coalgebra chapter, together with the material provided for the last three classes on coalgebra).
  • The deadline for submitting the sixth homework has been extended to January 6, 2014.
  • An example of a coinductive proof can be found here; the slides of the full lecture (in Dutch, for teachers and high school kids) are here.



  • Lecturers: Yde Venema (phone: 525 5299) and Luca Spada


  • Classes run from October 29 until December 11; there will be 14 classes in total.
  • There are two classes weekly:
    • on Tuesdays from 09.00 – 10.45 in room A1.14, and
    • on Wednesdays from 13.00 – 14.45 in room G3.13

    Both rooms are in Science Park.

  • Tuesday classes will be on Coalgebra, taught by Yde Venema, Wednesday classes will be on Algebra, taught by Luca Spada.

Course material

  • The basic course material for the course is the following text:
    Y Venema, Algebras and Coalgebras,
    in: J van Benthem, P Blackburn and F Wolter (editors), Handbook of Modal Logic,
    Elsevier, Amsterdam, 2006, pp 331-426.


  • Grading is through homework assignme and a final exam.

Course Description

Modal languages are simple yet expressive and flexible tools for describing all kinds of relational structures. Thus modal logic finds applications in many disciplines such as computer science, mathematics, linguistics or economics. Notwithstanding this enormous diversity in appearance and application area, modal logics have a great number of properties in common. This common mathematical backbone forms the topic of this course. This year the course will focus on algebraic and coalgebraic aspects of modal logic.


More specifically, we will cover (at least) the following topics:

  • Algebra: Boolean algebras, modal algebras, boolean algebras with operators, algebraizing modal logic, Lindenbaum-Tarski algebras, free algebras, complex/discrete duality, topologial duality, varieties of BAOs and their properties, canonicity.
  • Coalgebra: set functors and their coalgebras, final coalgebra, bisimilarity and behavioural equivalence, coinduction and covarieties, coalgebraic modal logic (both based on relation lifting and on predicate liftings), algebra and coalgebra.


It is assumed that students entering this class possess

  • a working knowledge of modal logic (roughly corresponding to the first sections of the Chapters 1-4 of the Modal Logic book by Blackburn, de Rijke and Venema).
  • some mathematical maturity.

Basic knowledge of general algebra, topology and category theory will be handy but not necessary.

Comments, complaints, questions: mail Yde Venema