Two isomorphism criteria
These are the slides of a presentation given in Stellenbosch on two isomorphisms criteria for co-limits of sequences of finitely presented or finitely generated objects in a locally small categories. The preprint containing full proofs and applications of these two results is available here.
Course on Algebra and Coalgebra
This is a back up reference of the page of the course on Algebra and Coalgebra (Autumn 2013).
Modal Logic, Algebra and Coalgebra (Autumn 2013)
Contents of these pages
News
- 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.
Practicalities
Staff
- Lecturers: Yde Venema (phone: 525 5299) and Luca Spada
Dates/location:
- 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
- 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.
Content
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.
Prerequisites
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.
Geometrical dualities for Łukasiewicz logic
This is the transcript of a featured talk given on the 15th of September 2011 at the XIX Congeresso dell’Unione Matematica Italiana held in Bologna, Italy. It is based on a joint work with Vincenzo Marra of the University of Milan that was published in Vincenzo Marra and Luca Spada. The dual adjunction between MV-algebras and Tychonoff spaces, Studia Logica 100(1-2):253-278, 2012. Special issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors).
The article develops a general dual adjunction between MV-algebras (the algebraic equivalents of Łukasiewicz logic) and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. Such a dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. Further the duality theorem for finitely presented objects is obtained from the general adjunction by a further specialisation. The treatment is aimed at emphasising the generality of the framework considered here in the prototypical case of MV-algebras.
Geometrical dualities for Łukasiewicz logic
Two isomorphism criteria for directed colimits
Using the general notions of finitely presentable and finitely generated object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally small) category, two sequences of finitely presentable objects and morphisms (or two sequences of finitely generated objects and monomorphisms) have isomorphic colimits (=direct limits) if, and only if, they are confluent. The latter means that the two given sequences can be connected by a back-and-forth chain of morphisms that is cofinal on each side, and commutes with the sequences at each finite stage. In several concrete situations, analogous isomorphism criteria are typically obtained by ad hoc arguments. The abstract results given here can play the useful rôle of discerning the general from the specific in situations of actual interest. We illustrate by applying them to varieties of algebras, on the one hand, and to dimension groups—the ordered $K_{0}$ of approximately finite-dimensional C*-algebras—on the other. The first application encompasses such classical examples as Kurosh’s isomorphism criterion for countable torsion-free Abelian groups of finite rank. The second application yields the Bratteli-Elliott Isomorphism Criterion for dimension groups. Finally, we discuss Bratteli’s original isomorphism criterion for approximately finite-dimensional C*-algebras, and show that his result does not follow from ours.
Two isomorphism criteria for directed colimits
Algebra|Coalgebra seminar
Starting from October 2013 I am organising the Algebra|Coalgebra seminar at the ILLC. The webpage of the seminar are here. See you there!
