# Mathematical Structures in Logic

#### Universiteit van Amsterdam

The final exam is now available at: Exam 2018. It is due on Monday, 9 April at 19:00. Good luck!

## Practicalities

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

• Teaching assistant: Saul Fernandez, email:saul.fdez.glez[at]gmail.com

• Time and Place: Lectures, Tuesday 9:00-11:00 and Thursday 13:00-15:00 (SP 107, F2.19); Tutorials, Tuesday 13:00-15:00 (SP 107, F2.19)

• EC: 6

• Assessment : There will be 6 or 7 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.

• General Topology, J. Kelley, 2008.

• General Topology, S. Willard, 2004.

• Topology, J. Munkres, 2000.

• Stone Spaces, P. Johnstone, 1986.

• ## Homeworksheets

• Homework 1 due on 13 February before the tutorial class.

• Homework 2 due on 20 February before the tutorial class.

• Homework 3 due on 27 February before the tutorial class.

• Homework 4 due on 6 March before the tutorial class.

• Homework 5 due on 13 March before the tutorial class.

• Homework 6 due on 20 March before the tutorial class.

• ## Lectures

 6 Feb 2018Tuesday Lecture9:00-11:00SP 107, F2.19 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). 6 Feb 2016Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 1. 8 Feb 2018Thursday Lecture 13:00-15:00SP 107, F2.19 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). 13 Feb 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 Representation of finite Boolean algebras (5.4-5.7 in Lat and Ord), Heyting algebras, equational definition, infinite distributive law, linear Heyting algebras. (See Section 2.2.1 in here.) 13 Feb 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 2. 15 Feb 2018Thursday Lecture 13:00-15:00SP 107, F2.19 Heyting algebras of open sets of a topological space, Alexandroff topologies, Heyting algebras of up-sets of a poset, interior algebras, topological insight on Goedel's embedding, Boolean algebra of regular open elements of a Heyting algebra, Glivenko's theorem. A short summary can be found here Notes 1 20 Feb 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 Glivenko's theorem (sketch of the main idea), topological insight into the Goedel translation, filters and ideals, maximal, prime and ultrafilters of Boolean algebras, Prime filter theorem (Sections 2.20-2.21, 10.7-10.18 in Lat and Ord). Note that in the lectures we worked mostly with filters, whereas Lat and Ord works mostly with ideals. 20 Feb 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 3. 22 Feb 2018Thursday Lecture 13:00-15:00SP 107, F2.19 Stone representation theorem (Sections 10.15-10.18, 10.20-10.22, 11.1-11.4 in Lat and Ord) 27 Feb 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 Stone duality, Congruences, homomorphic images, filters. (Sections 11.1-11.7). 27 Feb 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 4. 1 March Feb 2018Thursday Lecture 13:00-15:00SP 107, F2.19 Priestley duality. (Sections 11.17 - 11.32 in Lat & Ord). Note that in Lat and Ord Stone spaces are called Boolean spaces. 6 March 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 Esakia duality. Alexandroff and Stone-Cech compactifications. (Rough notes of the the material covered in the first 5 weeks is available on the Blackboard page of the course, consult also Morandi notes, Sections 3 -5.) 6 March 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 5. 8 March 2018Thursday Lecture 13:00-15:00SP 107, F2.19 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. 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. Jonsson's Lemma, finitely generated varieties. See the course notes on Blackboard. 13 March 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 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.) See also the notes on Blackboard. 13 March Feb 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 6. 15 March 2018Thursday Lecture 13:00-15:00SP 107, F2.19 Finitely generated varieties, finitely generated algebras, locally finite varieties, the Rieger-Nishimura lattice, locally finite varieties have the FMP, Kripke complete varieties, topologically complete varieties, the FMP of HA, the McKinsey-Tarski theorem. (See the notes on Blackboard.) 20 March 2018Tuesday Lecture 9:00-11:00SP 107, F2.19 Logics axiomatized by meet-implication formulas have the FMP, modal algebras, S4-algebras (also called closure algebras or interior algebras), duality for modal algebras, K4-algebras and S4-algebras, Esakia's lemma. See the notes on the Blackboard. 20 March Feb 2018Tuesday Tutorial 13:00-15:00SP 107, F2.19 The tutorial exercises can be found here TUT 7. 22 March 2018Thursday Lecture 13:00-15:00SP 107, F2.19 The connection of closure algebras and Heyting algebras, modal companions of intermediate logics, (Section 4.4 in Notes on intuitionistic logic). See also Modal companions. Sahlqvist completeness theorem (see the lecture notes on Blackboard).