Mathematical Structures in Logic

February - March 2016

Institute for Logic, Language and Computation

Universiteit van Amsterdam



A summary of the last few lectures is available! See the description of the last lecture.

The exam is now available here: Exam. It is due on Tuesday, 5 April at 14:00. Good luck!



Practicalities

  • Instructor: Nick Bezhanishvili, email: N.Bezhanishvili[at]uva.nl

  • Teaching assistant: Frederik Lauridsen, email: f.m.lauridsen[at]uva.nl

  • Time and Place: Lectures, Tuesday 9:00-11:00 (SP 904, B0. 203) and Thursday 13:00-15:00 (SP 904, B0. 207); Tutorials, Thursday 15:00-17:00 (SP 904, B0. 207)

  • EC: 6

  • Assessment : There will be 6 Homework sheets and a take home exam.

  • Exam: There will be a take home exam. The exam will be available after the last lecture and will be due after two weeks.


Study materials

  • A Course in Universal Algebra, Burris and Sankappanavar, 2012.

  • Introduction to Lattices and Order, Second Edition, Davey and Priestley, Cambridge University Press, 2002.

    • Additional literature

  • General Topology, J. Kelley, 2008.

  • General Topology, S. Willard, 2004.

  • Topology, J. Munkres, 2000.

  • Stone Spaces, P. Johnstone, 1986.

    • Homeworksheets

  • Homework 1, due: 9 February before class.

  • Homework 2, due: 16 February before class.

  • Homework 3, due: 23 February before class.

  • Homework 4, due: 1 March before class.

  • Homework 5, due: 8 March before class.

  • Homework 6, due: 15 March before class. For the exercise 3(a) consult slides 61, 65, 66 in Tutorial on varieties of Heyting algebras. Slides 65-66 have a sketch of the proof, but you need to fill the missing details.


    • Lectures

    2 Feb 2016
    Tuesday    		 Lecture
    9:00-11:00
    B0. 203    		 Definitions of lattices. The equivalence of the two definitions. Distributive lattices (Section 1-3, Ch. 1 in Univ. Alg. , 2.1-2.6, 2.8-2.14, 4.4, 4.10 in Lat and Ord. We did not prove 4.10).
    4 Feb 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Distributive and modular lattices, complete lattices, Boolean lattices and Boolean algebras (Section 3 in Univ Alg, we didn't prove Theorems 3.5 and 3.6., and Sections 4.13 - 4.18 in Lat and Ord).
    4 Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 1.
    9 Feb 2016
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Heyting algebras, equational definition, infinite distributive law, linear Heyting algebras, Heyting algebras of up-sets of a poset. (See Section 2.2.1 in here.)
    11 Feb 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Heyting algebras of open sets of a topological space, interior algebras, topological insight on Goedel's embedding, Boolean algebra of regular open elements of a Heyting algebra, Glivenko's theorem.
    11 Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 2.
    16 Feb 2016
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Congruences, homomorphic images, ideals and filters, maximal, prime and ultrafilters of Boolean algebras (Sections 6.1-6.10, 2.20-2.21, 10.7-10.12 in Lat and Ord). Note that in the lectures we worked mostly with filters, whereas Lat and Ord works mostly with ideals.
    18 Feb 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Prime filter theorem, Stone representation theorem (Sections 10.15-10.18, 10.20-10.22, 11.1-11.4 in Lat and Ord)
    18 Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 3.
    23 Feb 2016
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Stone duality (Sections 11.1-11.6).
    25 Feb 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Stone duality, Alexandroff and Stone-Cech compactifications of natural numbers, Priestley duality. (Sections 11.7 - 11.10, 11.17 - 11.27 in Lat & Ord). Note that in Lat and Ord Stone spaces are called Boolean spaces.
    25 Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 4.
    1 March 2015
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Priestley duality, Esakia spaces. (Sections 11.18 - 11.32 in Lat & Ord, check also the notes of Pat Morandi on duality in lattice theory, Sections 3 -5.)
    3 March 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Universal algebras, H, S and P (Ch 2. Sections 1-2, Univ. Alg.), subdirectly irreducible algebras, varieties (Ch 2. Sections 8-9, Univ. Alg.), Birkhoff's variety theorem. Morphisms between Esakia spaces, Esakia duality, the correspondence between congruences and closed sets for distributive latices, the correspondence between congruences and closed up-sets for Heyting algebras, Subdirectly irreducible Boolean algebras, distributive lattices and Heyting algebras. Sections 2.3.2 - 2.3.4 here, Sections 11.27 - 11.32 in Lat and Ord, consult also Morandi's notes.
    3 March 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 5.
    8 March 2015
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Algebraic completeness of classical and intuitionistic logics (Sections 11.11 - 11.16 in Lat & Ord, consult also Section 4.3 in Notes on intuitionistic logic, and slides 1-14 in Tutorial on varieties of Heyting algebras.)
    10 March 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Jonsson's Lemma, finitely generated varieties, finitely generated algebras, locally finite varieties, locally finite varieties have the FMP, the Rieger-Nishimura lattice, (see Theorem 6.8 and Corollary 6.10 in Univ Alg. , Sections 2.3.5, 3.1.1, 3.1.2 , 4.1.2)
    10 March Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 6.
    15 March 2015
    Tuesday    		 Lecture
    9:00-11:00
    SP B0.203    		 Logics axiomatized by meet-implication formulas have the FMP, canonical varieties and Kripke completeness, S4-algebras (also called closure algebras or interior algebras), the connection of closure algebras and Heyting algebras (Section 4.4 in Notes on intuitionistic logic).
    17 March 2016
    Thursday    		 Lecture
    13:00-15:00
    SP B0.207    		 Modal companions of intermediate logics (see slides 16-38 in Crash course on intermediate logics and modal companions). A short summary of the last 2-3 lectures can be found here Modal companions.
    17 March Feb 2016
    Thursday    		 Tutorial
    15:00-17:00
    SP B0.207    		 The tutorial exercises can be found here TUT 7.